Korovkin-type Theorems and Approximation by Positive Linear Operators
aa r X i v : . [ m a t h . C A ] S e p Korovkin-type Theorems and Approximation by Positive LinearOperators
Francesco Altomare
10 September 2010
To the memory of my parentsMaria Giordano (1915-1989) and Luigi Altomare (1898-1963)
Abstract
This survey paper contains a detailed self-contained introduction to Korovkin-type theoremsand to some of their applications concerning the approximation of continuous functions as wellas of L p -functions, by means of positive linear operators.The paper also contains several new results and applications. Moreover, the organization ofthe subject follows a simple and direct approach which quickly leads both to the main resultsof the theory and to some new ones.MSC: 41A36, 46E05, 47B65Keywords: Korovkin-type theorem, positive operator, approximation by positive operators,Stone-Weierstrass theorem, (weighted) continuous function space, L p -space. Contents C ( X ) C ( X ) , X compact 131 Surveys in Approximation TheoryVolume 6, 2010. pp. 92–164.c (cid:13) orovkin-type theorems and positive operators L p ( X, e µ ) spaces 1359 Korovkin-type theorems and Stone-Weierstrass theorems 14310 Korovkin-type theorems for positive projections 14611 Appendix: A short review of locally compact spaces and of some continuousfunction spaces on them 151References 157 Korovkin-type theorems furnish simple and useful tools for ascertaining whether a given sequence ofpositive linear operators, acting on some function space is an approximation process or, equivalently,converges strongly to the identity operator.Roughly speaking, these theorems exhibit a variety of test subsets of functions which guaranteethat the approximation (or the convergence) property holds on the whole space provided it holdson them.The custom of calling these kinds of results “Korovkin-type theorems” refers to P. P. Korovkinwho in 1953 discovered such a property for the functions , x and x in the space C ([0 , ,
1] as well as for the functions , cos and sin in the spaceof all continuous 2 π -periodic functions on the real line ([77-78]).After this discovery, several mathematicians have undertaken the program of extending Ko-rovkin’s theorems in many ways and to several settings, including function spaces, abstract Banachlattices, Banach algebras, Banach spaces and so on. Such developments delineated a theory whichis nowadays referred to as Korovkin-type approximation theory.This theory has fruitful connections with real analysis, functional analysis, harmonic analysis,measure theory and probability theory, summability theory and partial differential equations. Butthe foremost applications are concerned with constructive approximation theory which uses it as avaluable tool.Even today, the development of Korovkin-type approximation theory is far from complete,especially for those parts of it that concern limit operators different from the identity operator (seeProblems 5.3 and 5.4 and the subsequent remarks).A quite comprehensive picture of what has been achieved in this field until 1994 is documentedin the monographs of Altomare and Campiti ([8], see in particular Appendix D), Donner ([46]),Keimel and Roth ([76]), Lorentz, v. Golitschek and Makovoz ([83]). More recent results can befound, e.g., in [1], [9-15], [22], [47-52], [63], [71-74], [79], [114-116], [117] and the references therein.The main aim of this survey paper is to give a detailed self-contained introduction to the fieldas well as a secure entry into a theory that provides useful tools for understanding and unifyingseveral aspects pertaining, among others, to real and functional analysis and which leads to severalapplications in constructive approximation theory and numerical analysis.This paper, however, not only presents a survey on Korovkin-type theorems but also containsseveral new results and applications. Moreover, the organization of the subject follows a simple . Altomare R d , d ≥ C ( X ) of all real-valued continuous functions vanishing atinfinity on a locally compact space X and, in particular, in the space C ( X ) of all real-valuedcontinuous functions on a compact space X .We choose these continuous function spaces because they play a central role in the whole theoryand are the most useful for applications. Moreover, by means of them it is also possible to easilyobtain some Korovkin-type theorems in weighted continuous function spaces and in L p -spaces,1 ≤ p . These last aspects are treated at the end of Section 6 and in Section 8.We point out that we discuss Korovkin-type theorems not only with respect to the identityoperator but also with respect to a positive linear operator on C ( X ) opening the door to a varietyof problems some of which are still unsolved.In particular, in Section 10, we present some results concerning positive projections on C ( X ) , X compact, as well as their applications to the approximation of the solutions of Dirichlet problemsand of other similar problems.In Sections 6 and 7, we present several results and applications concerning Korovkin sets forthe identity operator. In particular, we show that, if M is a subset of C ( X ) that separates thepoints of X and if f ∈ C ( X ) is strictly positive, then { f } ∪ f M ∪ f M is a Korovkin set in C ( X ).This result is very useful because it furnishes a simple way to construct Korovkin sets, but inaddition, as we show in Section 9, it turns out that it is equivalent to the Stone generalization to C ( X )-spaces of the Weierstrass theorem. This equivalence was already established in [8, Section4.4] (see also [12-13]) but here we furnish a different, direct and more transparent proof.We also mention that, at the end of Sections 7 and 10, we present some applications concerningBernstein-Schnabl operators associated with a positive linear operator and, in particular, witha positive projection. These operators are useful for the approximation of not just continuousfunctions but also — and this was the real reason for the increasing interest in them — positivesemigroups and hence the solutions of initial-boundary value evolution problems. These aspectsare briefly sketched at the end of Section 10.Following the main aim of “Surveys in Approximation Theory”, this paper is directed to the orovkin-type theorems and positive operators R d , d ≥
1, or with an open or a closed subset of it or with the intersection of an open subsetand a closed subset of R d . However, this restriction does not produce any simplification of theproofs or of the methods.For the convenience of the reader and to make the exposition self-contained, we collect all theseprerequisites in the Appendix. There, the reader can also find some new simple and direct proofsof the main properties of Radon measures which are required throughout the paper, so that no apriori knowledge of the theory of Radon measures is needed.This paper contains introductory materials so that many aspects of the theory have been omit-ted. We refer, e.g., to [8, Appendix D] for further details about some of the main directionsdeveloped during the last fifty years.Furthermore, in the applications shown throughout the paper, we treat only general constructiveaspects (convergence of the approximation processes) without any mention of quantitative aspects(estimates of the rate of convergence, direct and converse results and so on) nor to shape preservingproperties. For such matters, we refer, e.g., to [8], [26], [38], [41], [42], [44], [45], [64], [65], [81-82],[83], [92], [109].We also refer to [19, Proposition 3.7], [67], [69] and [84] where other kinds of convergenceresults for sequences of positive linear operators can be found. The results of these last papers donot properly fall into the Korovkin-type approximation theory but they can be fruitfully used todecide whether a given sequence of positive linear operators is strongly convergent (not necessarilyto the identity operator).Finally we wish to express our gratitude to Mirella Cappelletti Montano, Vita Leonessa andIoan Ra¸sa for the careful reading of the manuscript and for many fruitful suggestions. We arealso indebted to Carl de Boor, Allan Pinkus and Vilmos Totik for their interest in this work aswell as for their valuable advice and for correcting several inaccuracies. Finally we want to thankMrs. Voichita Baraian for her precious collaboration in preparing the manuscript in LaTeX forfinal processing. In this section, we assemble the main notation which will be used throughout the paper togetherwith some generalities.Given a metric space (
X, d ), for every x ∈ X and r >
0, we denote by B ( x , r ) and B ′ ( x , r )the open ball and the closed ball with center x and radius r , respectively, i.e., B ( x , r ) := { x ∈ X | d ( x , x ) < r } (2.1)and B ′ ( x , r ) := { x ∈ X | d ( x , x ) ≤ r } . (2.2)The symbol F ( X ) . Altomare X . If M is a subset of F ( X ), thenby L ( M ) we designate the linear subspace generated by M . We denote by B ( X )the linear subspace of all functions f : X −→ R that are bounded, endowed with the norm ofuniform convergence (briefly, the sup-norm) defined by k f k ∞ := sup x ∈ X | f ( x ) | ( f ∈ B ( X )) , (2.3)with respect to which it is a Banach space.The symbols C ( X ) and C b ( X )denote the linear subspaces of all continuous (resp. continuous and bounded) functions in F ( X ).Finally, we denote by U C b ( X )the linear subspace of all uniformly continuous and bounded functions in F ( X ). Both C b ( X ) and U C b ( X ) are closed in B ( X ) and hence, endowed with the norm (2.3), they are Banach spaces.A linear subspace E of F ( X ) is said to be a lattice subspace if | f | ∈ E for every f ∈ E. (2.4)For instance, the spaces B ( X ) , C ( X ) , C b ( X ) and U C b ( X ) are lattice subspaces.Note that from (2.4), it follows that sup( f, g ) , inf( f, g ) ∈ E for every f, g ∈ E wheresup( f, g )( x ) := sup( f ( x ) , g ( x )) ( x ∈ X ) (2.5)and inf( f, g )( x ) := inf( f ( x ) , g ( x )) ( x ∈ X ) . (2.6)This follows at once by the elementary identitiessup( f, g ) = f + g + | f − g | f, g ) = f + g − | f − g | . (2.7)More generally, if f , . . . , f n ∈ E, n ≥
3, then sup ≤ i ≤ n f i , inf ≤ i ≤ n f i ∈ E. We say that a linear subspace E of F ( X ) is a subalgebra if f · g ∈ E for every f, g ∈ E (2.8)or, equivalently, if f ∈ E for every f ∈ E . In this case, if f ∈ E and n ≥
1, then f n ∈ E and hencefor every real polynomial Q ( x ) := α x + α x + · · · + α n x n ( x ∈ R ) vanishing at 0, the function Q ( f ) := α f + α f + · · · + α n f n (2.9)belongs to E as well. If E contains the constant functions, then P ( f ) ∈ E for every real polynomial P . orovkin-type theorems and positive operators C ([ a, b ]) is such anexample). However every closed subalgebra of C b ( X ) is a lattice subspace (see Lemma 9.1).Given a linear subspace E of F ( X ), a linear functional µ : E −→ R is said to be positive if µ ( f ) ≥ f ∈ E, f ≥ . (2.10)The simplest example of a positive linear functional is the so-called evaluation functional at a point a ∈ X defined by δ a ( f ) := f ( a ) ( f ∈ E ) . (2.11)If ( Y, d ′ ) is another metric space, we say that a linear operator T : E −→ F ( Y ) is positive if T ( f ) ≥ f ∈ E, f ≥ . (2.12)Every positive linear operator T : E −→ F ( Y ) gives rise to a family ( µ y ) y ∈ Y of positive linearfunctionals on E defined by µ y ( f ) := T ( f )( y ) ( f ∈ E ) . (2.13)Below, we state some elementary properties of both positive linear functionals and positive linearoperators.In what follows, the symbol F stands either for the field R or for a space F ( Y ), Y being anarbitrary metric space.Consider a linear subspace E of F ( X ) and a positive linear operator T : E −→ F . Then:(i) For every f, g ∈ E, f ≤ g, T ( f ) ≤ T ( g ) (2.14)(ii) If E is a lattice subspace, then | T ( f ) | ≤ T ( | f | ) for every f ∈ E. (2.15)(iii) (Cauchy-Schwarz inequality) If E is both a lattice subspace and a subalgebra, then T ( | f · g | ) ≤ p T ( f ) T ( g ) ( f, g ∈ E ) . (2.16)In particular, if ∈ E , then T ( | f | ) ≤ T ( ) T ( f ) ( f ∈ E ) . (2.17)(iv) If X is compact, ∈ E and F is either R or B ( Y ), then T is continuous and k T k = k T ( ) k . (2.18)Thus, if µ : E −→ R is a positive linear functional, then µ is continuous and k µ k = µ ( ). . Altomare Korovkin’s theorem provides a very useful and simple criterion for whether a given sequence ( L n ) n ≥ of positive linear operators on C ([0 , approximation process , i.e., L n ( f ) −→ f uniformlyon [0 ,
1] for every f ∈ C ([0 , e m ( t ) := t m (0 ≤ t ≤
1) (3.1)( m ≥ Theorem 3.1. (Korovkin ([77])) Let ( L n ) n ≥ be a sequence of positive linear operators from C ([0 , into F ([0 , such that for every g ∈ { , e , e } lim n →∞ L n ( g ) = g uniformly on [0 , . Then, for every f ∈ C ([0 , , lim n →∞ L n ( f ) = f uniformly on [0 , . Below, we present a more general result from which Theorem 3.1 immediately follows.For every x ∈ [0 ,
1] consider the auxiliary function d x ( t ) := | t − x | (0 ≤ t ≤ . (3.2)Then d x = e − xe + x and hence, if ( L n ) n ≥ is a sequence of positive linear operators satisfying the assumptions of The-orem 3.1, we get lim n →∞ L n ( d x )( x ) = 0 (3.3)uniformly with respect to x ∈ [0 , n ≥ L n ( d x ) = ( L n ( e ) − x ) + 2 x ( L n ( e ) − x ) + x ( L n ( ) − ) . After these preliminaries, the reader can easily realize that Theorem 3.1 is a particular case ofthe following more general result which, together with its modification (i.e., Theorem 3.5) as wellas the further consequences presented at the beginning of Section 4, should also be compared withthe simple but different methods of [79].Consider a metric space (
X, d ). Extending (3.2), for any x ∈ X we denote by d x ∈ C ( X ) thefunction d x ( y ) := d ( x, y ) ( y ∈ X ) . (3.4) Theorem 3.2.
Let ( X, d ) be a metric space and consider a lattice subspace E of F ( X ) containingthe constant functions and all the functions d x ( x ∈ X ). Let ( L n ) n ≥ be a sequence of positivelinear operators from E into F ( X ) and let Y be a subset of X such that(i) lim n →∞ L n ( ) = uniformly on Y ; orovkin-type theorems and positive operators (ii) lim n →∞ L n ( d x )( x ) = 0 uniformly with respect to x ∈ Y .Then for every f ∈ E ∩ U C b ( X )lim n →∞ L n ( f ) = f uniformly on Y. Proof.
Consider f ∈ E ∩ U C b ( X ) and ε >
0. Since f is uniformly continuous, there exists δ > | f ( x ) − f ( y ) | ≤ ε for every x, y ∈ X, d ( x, y ) ≤ δ. On the other hand, if d ( x, y ) ≥ δ , then | f ( x ) − f ( y ) | ≤ k f k ∞ ≤ k f k ∞ δ d ( x, y ) . Therefore, for x ∈ X fixed, we obtain | f − f ( x ) | ≤ k f k ∞ δ d x + ε and hence, for any n ≥ | L n ( f )( x ) − f ( x ) L n ( )( x ) | ≤ L n ( | f − f ( x ) | )( x ) ≤ k f k ∞ δ L n ( d x )( x ) + εL n ( )( x ) . We may now easily conclude that lim n →∞ L n ( f ) = f uniformly on Y because of the assumptions (i)and (ii). (cid:3) Theorem 3.2 has a natural generalization to completely regular spaces (for more details, werefer to [15]). Furthermore, the above proof can be adapted to show the next result.
Theorem 3.3.
Consider ( X, d ) and E ⊂ F ( X ) as in Theorem 3.2. Consider a sequence ( L n ) n ≥ of positive linear operators from E into F ( X ) and assume that for a given x ∈ X (i) lim n →∞ L n ( )( x ) = 1 ;(ii) lim n →∞ L n ( d x )( x ) = 0 .Then, for every bounded function f ∈ E that is continuous at x , lim n →∞ L n ( f )( x ) = f ( x ) . Adapting the proof of Theorem 3.2, we can show a further result. We first state a preliminarylemma.
Lemma 3.4.
Let ( X, d ) be a locally compact metric space. Then for every compact subset K of X and for every ε > , there exist < ε < ε and a compact subset K ε of X such that B ′ ( x, ε ) ⊂ K ε for every x ∈ K. . Altomare Proof.
Given x ∈ K , there exists 0 < ε ( x ) < ε such that B ′ ( x, ε ( x )) is compact. Since K ⊂ S x ∈ K B ( x, ε ( x ) / x , . . . , x p ∈ K such that K ⊂ p S i =1 B ( x i , ε ( x i ) / ε := min ≤ i ≤ p ε ( x i ) <ε and K ε := p S i =1 B ′ ( x i , ε ( x i )). Now, if x ∈ K and y ∈ X and if d ( x, y ) ≤ ε , then there exists an i ∈ { , . . . , p } such that d ( x, x i ) ≤ ε ( x i ) /
2, and hence d ( y, x i ) ≤ d ( y, x ) + d ( x, x i ) ≤ ε ( x i ) . Therefore y ∈ K ε . (cid:3) Theorem 3.5.
Let ( X, d ) be a locally compact metric space and consider a lattice subspace E of F ( X ) containing the constant function and all the functions d x ( x ∈ X ) . Let ( L n ) n ≥ be asequence of positive linear operators from E into F ( X ) and assume that(i) lim n →∞ L n ( ) = uniformly on compact subsets of X ;(ii) lim n →∞ L n ( d x )( x ) = 0 uniformly on compact subsets of X .Then, for every f ∈ E ∩ C b ( X ) , lim n →∞ L n ( f ) = f uniformly on compact subsets of X. Proof.
Fix f ∈ E ∩ C b ( X ) and consider a compact subset K of X . Given ε >
0, consider 0 < ε < ε and a compact subset K ε of X as in Lemma 3.4.Since f is uniformly continuous on K ε , there exists 0 < δ < ε such that | f ( x ) − f ( y ) | ≤ ε for every x, y ∈ K ε , d ( x, y ) ≤ δ. Given x ∈ K and y ∈ X , if d ( x, y ) ≤ δ , then y ∈ B ′ ( x, ε ) ⊂ K ε and hence | f ( x ) − f ( y ) | ≤ ε. If d ( x, y ) ≥ δ , then | f ( x ) − f ( y ) | ≤ k f k ∞ δ d ( x, y ) . Therefore, once again, | f − f ( x ) | ≤ k f k ∞ δ d x + ε so that, arguing as in the final part of the proof of Theorem 3.2, we conclude thatlim n →∞ L n ( f )( x ) = f ( x )uniformly with respect to x ∈ K . (cid:3) Theorem 3.1 was obtained by P. P. Korovkin in 1953 ([77], see also [78]). However, in [35],H. Bohman showed a result like Theorem 3.1 by considering sequences of positive linear operatorson C ([0 , L ( f )( x ) = X i ∈ I f ( a i ) ϕ i ( x ) (0 ≤ x ≤ , where ( a i ) i ∈ I is a finite family in [0 ,
1] and ϕ i ∈ C ([0 , i ∈ I ). Finally, we point out that thegerm of the same theorem can be also traced back to a paper by T. Popoviciu ([95]).Korovkin’s theorem 3.1 (often called Korovkin’s first theorem ) has many important appli-cations in the study of positive approximation processes in C ([0 , orovkin-type theorems and positive operators Bernstein operators on C ([0 , B n ( f )( x ) := n X k =0 f (cid:16) kn (cid:17)(cid:18) nk (cid:19) x k (1 − x ) n − k (3.5)( n ≥ , f ∈ C ([0 , , ≤ x ≤ B n ( f ) is a polynomial of degree not greater than n . Theywere introduced by S. N. Bernstein ([34]) to give the first constructive proof of the Weierstrassapproximation theorem (algebraic version) ([119]).Actually, we have that: Theorem 3.6.
For every f ∈ C ([0 , , lim n →∞ B n ( f ) = f uniformly on [0 , . Proof.
Each B n is a positive linear operator on C ([0 , n ≥ B n ( ) = , B n ( e ) = e and B n ( e ) = n − n e + 1 n e . Therefore, the result follows from Theorem 3.1. (cid:3)
The original proof of Bernstein’s Theorem 3.6 is based on probabilistic considerations (namely,on the weak law of large numbers). For a survey on Bernstein operators, we refer, e.g., to [82] (seealso [42] and [8]).Note that Theorem 3.6 furnishes a constructive proof of the Weierstrass approximation theorem[119] which we state below. (For a survey on many other alternative proofs of Weierstrass’ theorem,we refer, e.g., to [93-94].)
Theorem 3.7.
For every f ∈ C ([0 , , there exists a sequence of algebraic polynomials thatuniformly converges to f on [0 , . Using modern language, Theorem 3.7 can be restated as follows “The subalgebra of all algebraic polynomials is dense in C ([0 , with respect to the uniform norm”. By means of Theorem 3.6, we have seen that the Weierstrass approximation theorem can be ob-tained from Korovkin’s theorem.It seems to be not devoid of interest to point out that, from the Weierstrass theorem, it is possibleto obtain a special version of Korovkin’s theorem which involves only positive linear operators L n , n ≥
1, such that L n ( C ([0 , ⊂ B ([0 , n ≥
1. This special version will be referred toas the restricted version of Korovkin’s theorem.Theorem 3.8.
The restricted version of Korovkin’s theorem and Weierstrass’ Approximation The-orem are equivalent. . Altomare
Proof.
We have to furnish a proof of the restricted version of Korovkin’s theorem based solely onthe Weierstrass Theorem.Consider a sequence of positive linear operators ( L n ) n ≥ from C ([0 , B ([0 , n →∞ L n ( g ) = g uniformly on [0 ,
1] for every g ∈ { , e , e } . As in the proof of Theorem 3.1, wethen get lim n →∞ L n ( d x )( x ) = 0uniformly with respect to x ∈ [0 , m ≥ x, y ∈ [0 , | x m − y m | ≤ m | y − x | and hence, recalling the function e m ( x ) = x m (0 ≤ x ≤ | e m − y m | ≤ m | e − y | ( y ∈ [0 , . An application of the Cauchy-Schwarz inequality (2.16) implies, for any n ≥ y ∈ [0 , | L n ( e m ) − y m L n ( ) | ≤ mL n ( | e − y | ) ≤ m p L n ( ) p L n (( e − y ) ) = m p L n ( ) q L n ( d y ) . Therefore, lim n →∞ L n ( e m ) = e m uniformly on [0 ,
1] for any m ≥ n →∞ L n ( P ) = P uniformly on [0 ,
1] for every algebraic polynomial P on [0 , M := sup n ≥ k L n k = sup n ≥ k L n ( ) k < + ∞ andfixing f ∈ C ([0 , ε >
0, there exists an algebraic polynomial P on [0 ,
1] such that k f − P k ≤ ε ,and an integer r ∈ N such that k L n ( P ) − P k ≤ ε for every n ≥ r , so that k L n ( f ) − f k ≤ k L n ( f ) − L n ( P ) k + k L n ( P ) − P k + k P − f k≤ M k f − P k + k L n ( P ) − P k + k P − f k ≤ ( M + 2) ε. (cid:3) For another proof of Korovkin’s first theorem which involves Weierstrass theorem, see [117].It is well-known that there are ”trigonometric” versions of both Korovkin’s theorem and Weier-strass’ theorem (see Theorems 4.3 and 4.6). Also, these versions are equivalent (see Theorem 4.7).In the sequel, we shall also prove that the generalizations of these two theorems to compact and tolocally compact settings are equivalent as well (see Theorem 9.4).We proceed now to illustrate another application of Korovkin’s theorem that concerns theapproximation of functions in L p ([0 , , ≤ p < + ∞ , by means of positive linear operators. Notethat Bernstein operators are not suitable to approximate Lebesgue integrable functions (see, forinstance, [82, Section 1.9]).The space C ([0 , L p ([0 , k f k p := (cid:16) Z | f ( t ) | p d t (cid:17) /p ( f ∈ L p ([0 , k f k p ≤ k f k ∞ if f ∈ C ([0 , . (3.7) orovkin-type theorems and positive operators ,
1] is dense in L p ([0 , Kantorovich polynomials introduced by L. V. Kantorovich ([75]) furnish the first con-structive proof of the above mentioned density result. They are defined by K n ( f )( x ) := n X k =0 h ( n + 1) Z k +1 n +1 kn +1 f ( t ) d t i(cid:18) nk (cid:19) x k (1 − x ) n − k (3.8)for every n ≥ , f ∈ L p ([0 , , ≤ x ≤
1. Each K n ( f ) is a polynomial of degree not greater than n and every K n is a positive linear operator from L p ([0 , C ([0 , C ([0 , Theorem 3.9. If f ∈ C ([0 , , then lim n →∞ K n ( f ) = f uniformly on [0 , . Proof.
A direct calculation which involves the corresponding formulas for Bernstein operatorsgives for n ≥ K n ( ) = , K n ( e ) = nn + 1 e + 12( n + 1)and K n ( e ) = n ( n − n + 1) e + 2 n ( n + 1) e + 13( n + 1) . Therefore, the result follows at once from Korovkin’s Theorem 3.1. (cid:3)
Before showing a result similar to Theorem 3.9 for L p -functions, we need to recall some prop-erties of convex functions.Consider a real interval I of R . A function ϕ : I −→ R is said to be convex if ϕ ( αx + (1 − α ) y ) ≤ αϕ ( x ) + (1 − α ) ϕ ( y )for every x, y ∈ I and 0 ≤ α ≤
1. If I is open and ϕ is convex, then, for every finite family( x k ) ≤ k ≤ n in I and ( α k ) ≤ k ≤ n in [0 ,
1] such that n P k =1 α k = 1, ϕ (cid:16) n X k =1 α k x k (cid:17) ≤ n X k =1 α k ϕ ( x k )( Jensen’s inequality ).The function | t | p ( t ∈ R ), 1 ≤ p < ∞ , is convex. Given a probability space (Ω , F , µ ), an openinterval I of R and a µ -integrable function f : Ω −→ I , then Z Ω f d µ ∈ I. Furthermore, if ϕ : I −→ R is convex and ϕ ◦ f : Ω −→ R is µ -integrable, then . Altomare ϕ (cid:16) Z Ω f d µ (cid:17) ≤ Z Ω ϕ ◦ f d µ ( Integral Jensen inequality ).In particular, if f ∈ L p (Ω , µ ) ⊂ L (Ω , µ ), then (cid:12)(cid:12)(cid:12)(cid:12)Z f d µ (cid:12)(cid:12)(cid:12)(cid:12) p ≤ Z | f | p d µ. (3.9)(For more details see, e.g., [29, pp.18–21].)After these preliminaries, we now proceed to show the approximation property of ( K n ) n ≥ in L p ([0 , Theorem 3.10. If f ∈ L p ([0 , , ≤ p < + ∞ , then lim n →∞ K n ( f ) = f in L p ([0 , . Proof.
For every n ≥
1, denote by k K n k the operator norm of K n considered as an operator from L p ([0 , L p ([0 , M ≥ k K n k ≤ M forevery n ≥
1. After that, the result will follow immediately because, for a given ε >
0, there exists g ∈ C ([0 , k f − g k p ≤ ε and there exists ν ∈ N such that, for n ≥ ν , k K n ( g ) − g k ∞ ≤ ε so that k K n ( f ) − f k p ≤ M k f − g k p + k K n ( g ) − g k p + k g − f k p ≤ (2 + M ) ε. Now, in order to obtain the desired estimate, we shall use the convexity of the function | t | p on R and inequality (3.9).Given f ∈ L p ([0 , n ≥ ≤ k ≤ n , we have indeed ( n + 1) Z k +1 n +1 kn +1 | f ( t ) | d t ! p ≤ ( n + 1) Z k +1 n +1 kn +1 | f ( t ) | p d t and hence, for every x ∈ [0 , | K n ( f )( x ) | p ≤ n X k =0 (cid:18) nk (cid:19) x k (1 − x ) n − k h ( n + 1) Z k +1 n +1 kn +1 | f ( t ) | d t i p ≤ n X k =0 (cid:18) nk (cid:19) x k (1 − x ) n − k ( n + 1) Z k +1 n +1 kn +1 | f ( t ) | p d t. Therefore, Z | K n ( f )( x ) | p d x ≤ n X k =0 (cid:18) nk (cid:19)(cid:16) Z x k (1 − x ) n − k d x (cid:17)(cid:16) ( n + 1) Z k +1 n +1 kn +1 | f ( t ) | p d t (cid:17) . orovkin-type theorems and positive operators B ( u, v ) := Z t u − (1 − t ) v − d t ( u > , v > , it is not difficult to show that, for 0 ≤ k ≤ n , Z x k (1 − x ) n − k d x = B ( k + 1 , n − k + 1) = 1( n + 1) (cid:0) nk (cid:1) , and hence Z | K n ( f )( x ) | p d x ≤ n X k =0 Z k +1 n +1 kn +1 | f ( t ) | p d t = Z | f ( t ) | p d t. Thus, k K n ( f ) k p ≤ k f k p for every f ∈ L p ([0 , k K n k ≤ (cid:3) Remarks 3.11.
1. For every f ∈ L p ([0 , , ≤ p < + ∞ , it can also be shown thatlim n →∞ K n ( f ) = f almost everywhere on [0 , f ∈ C ([0 , n ≥ x ∈ [0 , B n +1 ( f ) ′ ( x ) = n X h =0 ( n + 1) h f (cid:16) h + 1 n + 1 (cid:17) − f (cid:16) hn + 1 (cid:17)i(cid:18) nh (cid:19) x h (1 − x ) n − h = K n ( f ′ )( x ) . Therefore, by Theorem 3.9, we infer thatlim n →∞ B n ( f ) ′ = f ′ uniformly on [0 , . (3.10)More generally, if f ∈ C ([0 , m ≥ ≤ k ≤ m ,lim n →∞ B n ( f ) ( k ) = f ( k ) uniformly on [0 ,
1] (3.11)([82, Section 1.8]).3. Another example of positive approximating operators on L p ([0 , , ≤ p < + ∞ , is furnishedby the Bernstein-Durrmeyer operators defined by D n ( f )( x ) := n X k =0 (cid:16) Z ( n + 1) (cid:18) nk (cid:19) t k (1 − t ) n − k f ( t ) d t (cid:17)(cid:18) nk (cid:19) x k (1 − x ) n − k (3.12)( f ∈ L p ([0 , , ≤ x ≤
1) (see [53], [40], [8, Section 5.3.8]).We also refer the interested reader to [16] where a generalization of Kantorovich operators isintroduced and studied. . Altomare
In this section, we shall consider the space R d , d ≥
1, endowed with the Euclidean norm k x k = (cid:16) d X i =1 x i (cid:17) / ( x = ( x i ) ≤ i ≤ d ∈ R d ) . (4.1)For every j = 1 , . . . , d , we shall denote by pr j : R d −→ R the j -th coordinate function which is defined by pr j ( x ) := x j ( x = ( x i ) ≤ i ≤ d ∈ R d ) . (4.2)By a common abuse of notation, if X is a subset of R d , the restriction of each pr j to X will beagain denoted by pr j . In this framework, for the functions d x ( x ∈ X ) defined by (3.4), we get d x = k x k − d X i =1 x i pr i + d X i =1 pr i . (4.3)Therefore, from Theorem 3.5, we then obtain Theorem 4.1.
Let X be a locally compact subset of R d , d ≥ , i.e., X is the intersection of anopen subset and a closed subset of R d (see Appendix). Consider a lattice subspace E of F ( X ) containing { , pr , . . . , pr d , d P i =1 pr i } and let ( L n ) n ≥ be a sequence of positive linear operators from E into F ( X ) such that for every g ∈ { , pr , . . . , pr d , d P i =1 pr i } lim n →∞ L n ( g ) = g uniformly on compact subsets of X. Then, for every f ∈ E ∩ C b ( X )lim n →∞ L n ( f ) = f uniformly on compact subsets of X. The special case of Theorem 4.1 when X is compact follows indeed from Theorem 3.2 and isworth being stated separately. It is due to Volkov ([118]). Theorem 4.2.
Let X be a compact subset of R d and consider a sequence ( L n ) n ≥ of positive linearoperators from C ( X ) into F ( X ) such that for every g ∈ { , pr , . . . , pr d , d P i =1 pr i } lim n →∞ L n ( g ) = g uniformly on X. Then for every f ∈ C ( X ) lim n →∞ L n ( f ) = f uniformly on X. orovkin-type theorems and positive operators X is contained in some sphere of R d , i.e., d P i =1 pr i is constant on X , then the testsubset in Theorem 4.2 reduces to { , pr , . . . , pr d } . (In [8, Corollary 4.5.2], the reader can find acomplete characterization of those subsets X of R d for which { , pr , . . . , pr d } satisfies Theorem4.2.)This remark applies in particular for the unit circle of R T := { ( x, y ) ∈ R | x + y = 1 } . (4.4)On the other hand, the space C ( T ) is isometrically (order) isomorphic to the space C π ( R ) := { f ∈ C ( R ) | f is 2 π -periodic } (4.5)(endowed with the sup-norm and pointwise ordering) by means of the isomorphism Φ : C ( T ) −→ C π ( R ) defined by Φ( F )( t ) := F (cos t, sin t ) ( t ∈ R ) . (4.6)Moreover, Φ( ) = , Φ( pr ) = cos , Φ( pr ) = sin (4.7)and so we obtain Korovkin’s second theorem . Theorem 4.3.
Let ( L n ) n ≥ be a sequence of positive linear operators from C π ( R ) into F ( R ) suchthat lim n →∞ L n ( g ) = g uniformly on R for every g ∈ { , cos , sin } . Then lim n →∞ L n ( f ) = f uniformly on R for every f ∈ C π ( R ) . Below, we discuss some applications of Theorem 4.3.For 1 ≤ p < + ∞ , we shall denote by L p π ( R )the Banach space of all (equivalence classes of) functions f : R −→ R that are Lebesgue integrableto the p -th power over [ − π, π ] and that satisfy f ( x + 2 π ) = f ( x ) for a.e. x ∈ R . The space L p π ( R )is endowed with the norm k f k p := (cid:16) π Z π − π | f ( t ) | p d t (cid:17) /p ( f ∈ L p π ( R )) . (4.8)A family ( ϕ n ) n ≥ in L π ( R ) is said to be a positive periodic kernel if every ϕ n is positive,i.e., ϕ n ≥ R , and lim n →∞ π Z π − π ϕ n ( t ) d t = 1 . (4.9) . Altomare ϕ n ) n ≥ generates a sequence of positive linear operators on L π ( R ). For every n ≥ , f ∈ L π ( R ) and x ∈ R , set L n ( f )( x ) := ( f ∗ ϕ n )( x ) = 12 π Z π − π f ( x − t ) ϕ n ( t ) d t (4.10)= 12 π Z π − π f ( t ) ϕ n ( x − t ) d t. From Fubini’s theorem and H¨older’s inequality, it follows that L n ( f ) ∈ L p π ( R ) if f ∈ L p π ( R ) , ≤ p < + ∞ .Moreover, if f ∈ C π ( R ), then the Lebesgue dominated convergence theorem implies that L n ( f ) ∈ C π ( R ). Furthermore, k L n ( f ) k p ≤ k ϕ n k k f k p ( f ∈ C π ( R )) (4.11)and k L n ( f ) k ∞ ≤ k ϕ n k k f k ∞ ( f ∈ C π ( R )) . (4.12)A positive kernel ( ϕ n ) n ≥ is called an approximate identity if for every δ ∈ ]0 , π [lim n →∞ Z − δ − π ϕ n ( t ) d t + Z πδ ϕ n ( t ) d t = 0 . (4.13) Theorem 4.4.
Consider a positive kernel ( ϕ n ) n ≥ in L π ( R ) and the corresponding sequence ( L n ) n ≥ of positive linear operators defined by (4.10). For every n ≥ , set β n := 12 π Z π − π ϕ n ( t ) sin t t. (4.14) Then the following properties are equivalent:a) For every ≤ p < + ∞ and f ∈ L p π ( R )lim n →∞ L n ( f ) = f in L p π ( R ) as well as lim n →∞ L n ( f ) = f in C π ( R ) provided f ∈ C π ( R ) .b) lim n →∞ β n = 0 .c) ( ϕ n ) n ≥ is an approximate identity. Proof.
To show the implication ( a ) ⇒ ( b ), it is sufficient to point out that for every n ≥ x ∈ R β n = 12 π Z π − π ϕ n ( u − x ) sin u − x u = 12 (cid:16) π Z π − π ϕ n ( t ) d t − (cos x ) L n (cos)( x ) − (sin x ) L n (sin)( x ) (cid:17) orovkin-type theorems and positive operators β n → n → ∞ .Now assume that (b) holds. Then, for 0 < δ < π and n ≥ ( δ/ π (cid:16) − δ Z − π ϕ n ( t ) d t + π Z δ ϕ n ( t ) d t (cid:17) ≤ π (cid:16) − δ Z − π ϕ n ( t ) sin t t + π Z δ ϕ n ( t ) sin t t (cid:17) ≤ β n and hence (c) follows.We now proceed to show the implication ( c ) ⇒ ( a ). Set M := sup n ≥ π R − π ϕ n ( t ) d t . For a given ε > δ ∈ ]0 , π [ such that | cos t − | ≤ ε M + 1) and | sin t | ≤ ε M + 1) for any t ∈ R , | t | ≤ δ ,and hence, for sufficiently large n ≥ (cid:12)(cid:12)(cid:12) π Z π − π ϕ n ( t ) d t − (cid:12)(cid:12)(cid:12) ≤ ε Z δ ≤| t |≤ π ϕ n ( t ) d t ≤ π ε . Therefore, for every x ∈ R , | L n (sin)( x ) − sin x | ≤ π π Z − π | sin( x − t ) − sin x | ϕ n ( t ) d t + (cid:12)(cid:12)(cid:12)(cid:16) π Z π − π ϕ n ( t ) d t − (cid:17)(cid:12)(cid:12)(cid:12) | sin x |≤ π Z δ ≤| t |≤ π | sin( x − t ) − sin x | ϕ n ( t ) d t + 12 π Z | t | <δ | (cos t −
1) sin x − cos x sin t | ϕ n ( t ) d t + ε/ ≤ π Z δ ≤| t |≤ π ϕ n ( t ) d t + ε M + 1) 12 π π Z − π ϕ n ( t ) d t + ε ≤ ε. Therefore, lim n →∞ L n (sin) = sin uniformly on R . The same method can be used to show thatlim n →∞ L n (cos) = cos uniformly on R and hence, by Korovkin’s second theorem 4.3, we obtainlim n →∞ L n ( f ) = f in C π ( R ) for every f ∈ C π ( R ).By reasoning as in the proof of Theorem 3.10, it is a simple matter to get the desired convergenceformula in L p π ( R ) by using the previous one on C π ( R ), the denseness of C π ( R ) in L p π ( R ) andformula (4.11) which shows that the operators L n , n ≥
1, are equibounded from L p π ( R ) into L p π ( R ). (cid:3) Two simple applications of Theorem 4.4 are particularly worthy of mention. For other applica-tions, we refer to [8, Section 5.4], [38], [41], [78]. . Altomare n ∈ N is a real function of theform u n ( x ) = 12 a + n X k =1 a k cos kx + b k sin kx ( x ∈ R ) (4.15)where a , a , . . . , a n , b , . . . , b n ∈ R . A series of the form12 a + ∞ X k =1 a k cos kx + b k sin kx ( x ∈ R ) (4.16)( a k , b k ∈ R ) is called a trigonometric series .If f ∈ L π ( R ), the trigonometric series12 a ( f ) + ∞ X k =1 a k ( f ) cos kx + b k ( f ) sin kx ( x ∈ R ) (4.17)where a ( f ) := 1 π Z π − π f ( t ) d t, (4.18) a k ( f ) := 1 π Z π − π f ( t ) cos kt d t, k ≥ , (4.19) b k ( f ) := 1 π Z π − π f ( t ) sin kt d t, k ≥ , (4.20)is called the Fourier series of f . The a n ’s and b n ’s are called the real Fourier coefficients of f .For any n ∈ N , denote by S n ( f )the n -th partial sum of the Fourier series of f , i.e., S ( f ) = 12 a ( f ) (4.21)and, for n ≥ S n ( f )( x ) = 12 a ( f ) + n X k =1 a k ( f ) cos kx + b k ( f ) sin kx. (4.22)Each S n ( f ) is a trigonometric polynomial; moreover, considering the functions D n ( t ) := 1 + 2 n X k =1 cos kt ( t ∈ R ) , (4.23)we also get S n ( f )( x ) = 12 π Z π − π f ( t ) D n ( x − t ) d t ( x ∈ R ) . (4.24)The function D n is called the n -th Dirichlet kernel . orovkin-type theorems and positive operators t/
2, we obtainsin t D n ( t ) = sin t n X k =1 sin (cid:16) k t (cid:17) − sin (cid:16) k − t (cid:17) = sin 1 + 2 n t, so that D n ( t ) = sin(1 + 2 n ) t/ t/ t is not a multiple of π, n + 1 if t is a multiple of π. (4.25) D n is not positive and ( D n ) n ≥ is not an approximate identity ([38, Prop. 1.2.3]). Moreover, thereexists f ∈ C π ( R ) such that ( S n ( f )) n ≥ does not converge uniformly (nor pointwise) to f , i.e., theFourier series of f does not converge uniformly (nor pointwise) to f .For every n ∈ N , put F n ( f ) := 1 n + 1 n X k =0 S k ( f ) . (4.26) F n ( f ) is a trigonometric polynomial. Moreover, from the identity(sin t n − X k =0 sin 2 k + 12 t = sin n t ( t ∈ R ) , (4.27)it follows that for every x ∈ R F n ( f )( x ) = 12 π Z π − π f ( t ) 1( n + 1) n X k =0 sin((2 k + 1)( x − t ) / x − t ) /
2) d t (4.28)= 12 π Z π − π f ( t ) 1( n + 1) sin (( n + 1)( x − t ) / (( x − t ) /
2) d t = 12 π Z π − π f ( t ) ϕ n ( x − t ) d t, where ϕ n ( x ) := ( sin (( n +1) x/ n +1) sin ( x/ if x is not a multiple of 2 π,n + 1 if x is a multiple of 2 π. (4.29)Actually, the sequence ( ϕ n ) n ≥ is a positive kernel which is called the Fej´er kernel , and thecorresponding operators F n , n ≥
1, are called the
Fej´er convolution operators . Theorem 4.5.
For every f ∈ L p π ( R ) , ≤ p < + ∞ , lim n →∞ F n ( f ) = f in L p π ( R ) and, if f ∈ C π ( R ) , lim n →∞ F n ( f ) = f in C π ( R ) . . Altomare Proof.
Evaluating the Fourier coefficients of , cos and sin, and by using (4.26), we obtain, for n ≥ F n ( ) = , F n (cos) = nn + 1 cos , F n (sin) = nn + 1 sinand β n = 12( n + 1) . The result now follows from Theorem 4.4 or, more directly, from Theorem 4.3. (cid:3)
Theorem 4.5 is due to Fej´er ([56-57]). It furnishes the first constructive proof of the Weierstrassapproximation theorem for periodic functions . Theorem 4.6. If f ∈ L p π ( R ) , ≤ p < + ∞ (resp. f ∈ C π ( R ) ) then there exists a sequence oftrigonometric polynomials that converges to f in L p π ( R ) (resp. uniformly on R ). As in the “algebraic” case, we shall now show that from Weierstrass’ approximation theorem,it is possible to deduce a “restricted” version of Theorem 4.3, where in addition it is required thateach operator L n maps C π ( R ) into B ( R ). We shall also refer to this version as the restrictedversion of Korovkin’s second theorem . Theorem 4.7.
The restricted version of Korovkin’s second Theorem 4.3 and Weierstrass’ Theorem4.6 are equivalent.
Proof.
An inspection of the proof of Theorem 4.5 shows that Theorem 4.3 implies Theorem 4.5and, hence, Theorem 4.6.Conversely, assume that Theorem 4.6 is true and consider a sequence ( L n ) n ≥ of positive linearoperators from C π ( R ) into B ( R ) such that L n ( g ) −→ g uniformly on R for every g ∈ { , cos , sin } .For every m ≥
1, set f m ( x ) := cos mx and g m ( x ) := sin mx ( x ∈ R ). Since the subspace of alltrigonometric polynomials is dense in C π ( R ) andsup n ≥ k L n k = sup n ≥ k L n ( ) k < + ∞ , it is enough to show that L n ( f m ) → f m and L n ( g m ) → g m uniformly on R for every m ≥ x ∈ R , consider the function Φ x ( y ) = sin x − y ( y ∈ R ). ThenΦ ( y ) := sin x − y − cos x cos y − sin x sin y ) ( y ∈ R )and hence L n (Φ x )( x ) → x ∈ R . On the other hand, for m ≥ x, y ∈ R , we get | f m ( x ) − f m ( y ) | = 2 (cid:12)(cid:12)(cid:12) sin m (cid:16) x + y (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sin m (cid:16) x − y (cid:17)(cid:12)(cid:12)(cid:12) ≤ c m (cid:12)(cid:12)(cid:12) sin (cid:16) x − y (cid:17)(cid:12)(cid:12)(cid:12) where c m := 2 sup α ∈ R sin mα sin α , and hence | f m ( x ) L n ( ) − L n ( f m ) | ≤ c m L n ( | Φ x | ) ≤ c m p L n ( ) p L n (Φ x ) . Therefore, L n ( f m )( x ) − f m ( x ) → x ∈ R . orovkin-type theorems and positive operators g m , m ≥
1, because | g m ( x ) − g m ( y ) | = 2 (cid:12)(cid:12)(cid:12) cos m (cid:16) x + y (cid:17) sin m (cid:16) x − y (cid:17)(cid:12)(cid:12)(cid:12) ≤ K m (cid:12)(cid:12)(cid:12) sin (cid:16) x − y (cid:17)(cid:12)(cid:12)(cid:12) and this completes the proof. (cid:3) For another short proof of Korovkin’s second theorem that uses the trigonometric version ofWeierstrass’ theorem, see [117].Fej´er’s Theorem 4.5 is noteworthy because it reveals an important property of the Fourier series.Actually, it shows that Fourier series are always
Cesaro-summable to f in L p π ( R ) or in C π ( R ),provided that f ∈ L p π ( R ) or f ∈ C π ( R ). Another deeper theorem, ascribed to Fej´er and Lebesgue,states that, if f ∈ L π ( R ), then its Fourier series is Cesaro-summable to f a.e. on R ([112, Theorem8.35]).Below, we further discuss another regular summation method, namely the Abel-summationmethod , which applies to Fourier series.We begin with the following equality1 + z − z = 1 + 2 ∞ X k =1 z k ( z ∈ C , | z | <
1) (4.30)which holds uniformly on any compact subset of { z ∈ C | | z | < } . Given x ∈ R and 0 ≤ r <
1, byapplying (4.30) to z = r e i x and by taking the real parts of both sides, we get1 + 2 ∞ X k =1 r k cos kx = 1 − r − r cos x + r (4.31)and the identity holds uniformly when r ranges in a compact interval of [0 , P r ( t ) := 1 − r − r cos t + r ( t ∈ R ) (4.32)(0 ≤ r <
1) is called the
Abel-Poisson kernel and the corresponding operators P r ( f )( x ) := 1 − r π Z π − π f ( t )1 − r cos( x − t ) + r d t ( x ∈ R ) (4.33)(0 ≤ r < , f ∈ L π ( R )) are called the Abel-Poisson convolution operators . Taking (4.31)into account, it is not difficult to show that P r ( f )( x ) = 12 a ( f ) + ∞ X k =1 r k ( a k ( f ) cos kx + b k ( f ) sin kx ) (4.34)where the coefficients a k ( f ) and b k ( f ) are the Fourier coefficients of f defined by (4.18)–(4.20). Theorem 4.8. If f ∈ L p π ( R ) , ≤ p < + ∞ , then lim r → − P r ( f ) = f in L p π ( R ) and, if f ∈ C π ( R ) , lim r → − P r ( f ) = f uniformly on R . . Altomare Proof.
The kernels p r , 0 ≤ r <
1, are positive. Moreover, from (4.34), we get P r ( ) = , P r (cos) = r cos , P r (sin) = r sinand hence β r = 1 − r . Therefore, the result follows from Theorem 4.4 (or from Theorem 4.5). (cid:3)
According to (4.34), Theorem 4.8 claims that the Fourier series of a function f ∈ L p π ( R ) (resp. f ∈ C π ( R )) is Abel summable to f in L p π ( R ) (resp. uniformly on R ).For further applications of Korovkin’s second theorem to approximation by convolution opera-tors and summation processes, we refer, e.g., to [8, Section 5.4], [38], [41], [78].We finally point out the relevance of Theorem 4.8 in the study of the Dirichlet problem forthe unit disk D := { ( x, y ) ∈ R | x + y ≤ } . Given F ∈ C ( ∂ D ) = C ( T ) ≡ C π ( R ), this problemconsists in finding a function U ∈ C ( D ) possessing second partial derivatives on the interior of D such that ∂ U∂x ( x, y ) + ∂ U∂y ( x, y ) = 0 ( x + y < ,U ( x, y ) = F ( x, y ) ( x + y = 1) . (4.35)By using polar coordinates x = r cos θ and y = r sin θ (0 ≤ r < , θ ∈ R ) and the functions f ( θ ) := F (cos θ, sin θ ) ( θ ∈ R ) and u ( r, θ ) := U ( r cos θ, r sin θ ) (0 ≤ r < , θ ∈ R ), problem (4.35)turns into ∂ u∂r ( r, θ ) + 1 r ∂u∂r ( r, θ ) + 1 r ∂ u∂θ ( r, θ ) = 0 0 < r < , θ ∈ R ,u (0 , θ ) = π π R − π f ( t ) d t θ ∈ R , lim r → − u ( r, θ ) = f ( θ ) uniformly w.r.t. θ ∈ R . (4.36). With the help of Theorem 4.8 it is not difficult to show that a solution to problem (4.36) isgiven by u ( r, θ ) = P r ( f )( θ ) (0 ≤ r < , θ ∈ R ) . (4.37)Accordingly, the function U ( x, y ) := (cid:26) u ( r, θ ) if x = r cos θ, y = r sin θ and x + y < ,F ( x, y ) if x + y = 1 , is a solution to problem (4.35). Furthermore, as it is well-known, the solution to (4.35) is unique. Asimilar result also holds if F : T → R is a Borel-measurable function such that π R π − π | F (cos t, sin t ) | p dt < + ∞ , ≤ p < + ∞ . For more details we refer, e.g., to [38, Proposition 1.2.10 and Theorem7.1.3] and to [96, Section 1.2].We end this section by presenting an application of Theorem 4.1. Consider the d -dimensionalsimplex K d := { x = ( x i ) ≤ i ≤ d ∈ R d | x i ≥ , ≤ i ≤ d, and d X i =1 x i ≤ } (4.38) orovkin-type theorems and positive operators n ≥ , f ∈ C ( K d ) and x = ( x i ) ≤ i ≤ d ∈ K d , set B n ( f )( x ) : = X h ,...,h d =0 ,...,nh + ··· + h d ≤ n f (cid:16) h n , . . . , h d n (cid:17) n ! h ! · · · h d !( n − h − · · · − h d )! × x h · · · x h d d (1 − x − · · · − x d ) n − h −···− h d . (4.39) B n ( f ) is a polynomial and it is usually called the n -th Bernstein polynomial on the d -dimensional simplex associated with f . These polynomials were first studied by Dinghas ([43]). Theorem 4.9.
For every f ∈ C ( K d ) , lim n →∞ B n ( f ) = f uniformly on K d . Proof.
From the multinomial theorem, it follows that B n ( ) = . Consider now the first coordinatefunction pr (see (4.2)). Then, for x = ( x i ) ≤ i ≤ d ∈ K d and n ≥ B n ( pr )( x ) = X h ,...,h d =0 ,...,nh + ··· + h d ≤ n h n n ! h ! · · · h d !( n − h − · · · − h d )! × x h · · · x h d d (1 − x − · · · − x d ) n − h −···− h d = x X k ,h ,...,h d =0 ,...,nk + h + ··· + h d ≤ n − ( n − k ! h ! · · · h d !(( n − − k − h − · · · − h d )! × x k x h · · · x h d d (1 − x − · · · − x d ) ( n − − k − h −···− h d = x . On the other hand, for n ≥ ,B n ( pr )( x ) = X h ,...,h d =0 ,...,nh + h + ··· + h d ≤ n h n n ! h ! · · · h d !( n − h − · · · − h d )! × x h · · · x h d d (1 − x − · · · − x d ) n − h −···− h d = X h + ··· + h d ≤ n h n ( n − h − h ! · · · h d !( n − h − · · · − h d )! × x h · · · x h d d (1 − x − · · · − x d ) n − h −···− h d = X h + ··· + h d ≤ n n − n h − n − n − h − h ! · · · h d !( n − h − · · · − h d )! × x h · · · x h d d (1 − x − · · · − x d ) n − h −···− h d + x n = n − n x + x n . Similar considerations apply to the other coordinate functions. Thus for every j = 1 , . . . , d , B n ( pr j ) = pr j and B n ( pr j ) = n − n pr j + 1 n pr j . (4.40) . Altomare B n ) n ≥ , we get the result. (cid:3) Remark 4.10.
Theorem 4.9 gives a constructive proof of the multidimensional versions of theWeierstrass approximation theorem, namely
For every f ∈ C ( K d ) , there exists a sequence of algebraic polynomials that converges to f uniformly on K d . Another useful generalization of the one-dimensional Bernstein polynomial is discussed below.Consider the hypercube Q d = [0 , d and for every n ≥ , f ∈ C ( Q d ) and x = ( x i ) ≤ i ≤ d , set B n ( f )( x ) := n X h ,...,h d =0 f (cid:16) h n , . . . , h d n (cid:17)(cid:18) nh (cid:19) · · · (cid:18) nh d (cid:19) x h (1 − x ) n − h · · · x h d d (1 − x d ) n − h d . (4.41)The polynomials B n ( f ), n ≥
1, are called the
Bernstein polynomials on the hypercube associated with f . They were first studied by Hildebrandt and Schoenberg ([70]) and Butzer ([37]).The proof of Theorem 4.9 works also for these operators giving the same formula (4.40). Therefore,again by Theorem 4.2, we obtain Theorem 4.11.
For every f ∈ C ( Q d ) , lim n →∞ B n ( f ) = f uniformly on Q d . We end the section by discussing some applications of Theorem 4.1 for noncompact subsets of R d .The first application is concerned with the interval [0 , + ∞ [ . Set E := { f ∈ C ([0 , + ∞ [) | there exist α ≥ M ≥ | f ( x ) | ≤ M exp( αx ) ( x ≥ } , (4.42)and for every f ∈ E, n ≥ , and x ≥
0, define M n ( f )( x ) := exp( − nx ) ∞ X k =0 f (cid:16) kn (cid:17) n k x k k ! . (4.43)The operator M n is linear and positive and is called the n -th Sz´asz-Mirakjan operator
Thesequence ( M n ) n ≥ was first introduced and studied by Mirakjan ([86]), Favard ([55]) and Sz´asz([113]) and is one of the most studied sequences of positive linear operators on function spaces on[0 , + ∞ [.A simple calculation shows that M n ( ) = , M n ( e ) = e and M n ( e ) = e + 1 n e . Therefore, from Theorem 4.1, we get that orovkin-type theorems and positive operators
Theorem 4.12.
For every f ∈ C b ([0 , + ∞ [) , lim n →∞ M n ( f ) = f uniformly on compact subsets of [0 , + ∞ [ . The next application concerns the case X = R d , d ≥
1. For every n ≥ f : R d → R , set G n ( f )( x ) := (cid:16) n π (cid:17) d/ Z R d f ( t ) exp( − n k t − x k ) d t (4.44)( x ∈ R d ).We shall consider the operators G n , n ≥
1, defined on the lattice subspace of all Borel-measurable functions f ∈ F ( R d ) for which the integral (4.44) is absolutely convergent. Theyare referred to as the Gauss-Weierstrass convolution operators on R d . Among other things,they are involved in the study of the heat equation on R d (see, e.g., [8], [38]).By using the following formulae: (cid:16) πσ (cid:17) / Z R exp (cid:16) − ( t − α ) σ (cid:17) d t = 1 , (4.45) (cid:16) πσ (cid:17) / Z R t exp (cid:16) − ( t − α ) σ (cid:17) d t = α, (4.46)and (cid:16) πσ (cid:17) / Z R t exp (cid:16) − ( t − α ) σ (cid:17) d t = α + σ , (4.47)( α ∈ R , σ >
0) (see [30, formulae (4.14) and (4.15)]) and appealing to Fubini’s theorem, weimmediately obtain for every i = 1 , . . . , d and x ∈ R d , G n ( )( x ) := d Y j =1 (cid:16) n π (cid:17) / Z R exp( − n t j − x j ) ) d t j = 1 , (4.48) G n ( pr i )( x ) = d Y j =1 j = i (cid:16) n π (cid:17) / Z R exp (cid:16) − n t j − x j ) (cid:17) d t j × (4.49) × (cid:16) n π (cid:17) / Z R t i exp (cid:16) − n t i − x i ) (cid:17) d t i = x i = pr i ( x )and G n ( pr i )( x ) = d Y j =1 j = i (cid:16) n π (cid:17) / Z R exp (cid:16) − n t j − x j ) (cid:17) d t j × (4.50) × (cid:16) n π (cid:17) / Z R t i exp (cid:16) − n t i − x i ) (cid:17) d t i = x i + 2 n = pr i ( x ) + 2 n . From Theorem 4.1, we then obtain . Altomare
Theorem 4.13.
For every f ∈ C b ( R d ) , lim n →∞ G n ( f ) = f uniformly on compact subsets of R d . For additional properties of Gauss-Weierstrass operators, we refer, e.g., to [38] (see also a recentgeneralization given in [21-23]).
After the discovery of Korovkin’s theorem, several mathematicians tried to extend it in severaldirections with the aim, for instance:(i) to find other subsets of functions satisfying the same property as { , e , e } ;(ii) to establish results like Theorem 3.1 in other function spaces or in abstract Banach spaces;(iii) to establish results like Theorem 3.1 for other classes of linear operators.As a consequence of these investigations, a new theory was created which is nowadays called“Korovkin-type Approximation Theory”. This theory has strong and fruitful connections not onlywith classical approximation theory but also with real analysis, functional analysis, harmonic anal-ysis, probability theory and partial differential equations. We refer to [8] for a rather detaileddescription of the development of this theory.We shall next discuss some of the main results of the theory obtained in the framework of thespaces C ( X ) ( X locally compact noncompact space), C ( X ) ( X compact space), occasionally in L p ( X, e µ ) spaces, 1 ≤ p < + ∞ , and in weighted continuous function spaces. These spaces play acentral role in the theory and they are the most useful in applications.In addition, it transpires that the elementary methods used in the previous sections are notappropriate to give a wider possibility to determine other test functions (like { , e , e } in C ([0 , R d , d ≥
1, and their open or closed subsets, many other topological spaces which are important intheir own right. The reader who is not interested in this level of generality, may replace everywhereour locally compact spaces with a space R d , d ≥
1, or with an open or a closed subset of it, or withthe intersection of an open subset and a closed subset of R d . However, this restriction does notproduce any simplification in the proofs or in the methods.In the sequel, given a locally compact Hausdorff space X , we shall denote by K ( X )the linear subspace of all real-valued continuous functions on X having compact support. Then K ( X ) ⊂ C b ( X ). We shall denote by C ( X )the closure of K ( X ) with respect to the sup-norm k · k ∞ (see (2.3)). Thus, C ( X ) is a closedsubspace of C b ( X ) and hence, endowed with the norm k · k ∞ , is a Banach space. orovkin-type theorems and positive operators X is compact, then C ( X ) = C ( X ). If X is not compact, then a function f ∈ C ( X ) belongsto C ( X ) if and only if for every ε > there exists a compact subset K of X such that | f ( x ) | ≤ ε for every x ∈ X \ K. For additional topological and analytical properties of locally compact spaces and of some relevantcontinuous function spaces on them, we refer to the Appendix.The spaces C ( X ) and C ( X ), ( X compact), endowed with the natural pointwise ordering andthe sup-norm, become Banach lattices. Similarly L p ( X, e µ ), endowed with the natural norm k · k p and the ordering f ≤ g if f ( x ) ≤ g ( x ) for e µ -a.e. x ∈ X, is a Banach lattice (for more details on ( L p ( X, e µ ), k · k p ) spaces, we refer to the Appendix (formulae(11.11) and (11.12))).For the reader’s convenience, we recall that a Banach lattice E is a vector space endowedwith a norm k · k and an ordering ≤ on E such that(i) ( E, k · k ) is a Banach space;(ii) ( E, ≤ ) is a vector lattice;(iii) If f, g ∈ E and | f | ≤ | g | then k f k ≤ k g k .(where | f | := sup( f, − f ) for every f ∈ E ).Actually it is convenient to state the main definitions of the theory in the framework of Banachlattices. However, the reader not accustomed to this terminology may replace our abstract spaceswith the concrete ones such as C ( X ), C ( X ) or L p ( X, e µ ). For more details on Banach lattices, werefer, e.g., to [2].If E and F are Banach lattices, a linear operator L : E −→ F is said to be positive if L ( f ) ≥ f ∈ E, f ≥ . Every positive linear operator L : E −→ F is continuous, (see, e.g., [2, Theorem 12.3]). Moreover,if E = C ( X ), X compact, then k L k = k L ( ) k .A lattice homomorphism S : E −→ F is a linear operator satisfying | S ( f ) | = S ( | f | ) forevery f ∈ E . Equivalently, this means that S preserves the finite lattice operation, i.e., for every f , . . . , f n ∈ E, n ≥ S (cid:18) inf ≤ i ≤ n f i (cid:19) = inf ≤ i ≤ n S ( f i ) and S (cid:18) sup ≤ i ≤ n f i (cid:19) = sup ≤ i ≤ n S ( f i ).For instance, if X is a locally compact Hausdorff space, e µ a regular finite Borel measure on X and1 ≤ p < + ∞ , then the natural embedding J p : C ( X ) −→ L p ( X, e µ ) defined by J p ( f ) := f ( f ∈ C ( X )) is a lattice homomorphism.Analogously, if X and Y are compact spaces and ϕ : Y −→ X is a continuous mapping, then thecomposition operator T ϕ ( f ) := f ◦ ϕ, ( f ∈ C ( X )) , is a lattice homomorphism from C ( X ) into C ( Y ).Every lattice homomorphism is positive and hence continuous. A linear bijection S : E −→ F is alattice homomorphism if and only if S and its inverse S − are both positive [2, Theorem 7.3]. In . Altomare S is a lattice isomorphism . When there exists a lattice isomorphismbetween E and F , then we say that E and F are lattice isomorphic .The following definition, which is one of the most important of the theory, is clearly motivatedby the Korovkin theorem and was first formulated by V. A. Baskakov ([28]). Definition 5.1.
A subset M of a Banach lattice E is said to be a Korovkin subset of E if forevery sequence ( L n ) n ≥ of positive linear operators from E into E satisfying(i) sup n ≥ k L n k < + ∞ ,and(ii) lim n →∞ L n ( g ) = g for every g ∈ M ,it turns out that lim n →∞ L n ( f ) = f for every f ∈ E. Note that, if E = C ( X ), X compact space, and the constant function belongs to the linearsubspace L ( M ) generated by M , then condition (i) is superfluous because it is a consequence of(ii).According to Definition 5.1, we may restate Korovkin’s Theorem 3.1 by saying that { , e , e } is a Korovkin set in C ([0 , M is a Korovkin subset of E if and only if the linear subspace L ( M ) generated by M is a Korovkin subset. In the sequel, a linear subspace that is a Korovkinsubset will be referred to as a Korovkin subspace of E . If E and F are lattice isomorphic and if S : E −→ F is a lattice isomorphism, then a subset M of E is a Korovkin subset in E if and onlyif S ( M ) is a Korovkin subset in F .Korovkin sets (when they exist) are useful for investigating the convergence of equiboundedsequences of positive linear operators towards the identity operator or, from the point of view ofapproximation theory, the approximation of every element f ∈ E by means of ( L n ( f )) n ≥ .According to Lorentz ([80]), who first proposed a possible generalization, it seems to be equallyinteresting to study the following more general concept. Definition 5.2.
Let E and F be Banach lattices and consider a positive linear operator T : E −→ F . A subset M of E is said to be a Korovkin subset of E for T if for every sequence ( L n ) n ≥ ofpositive linear operators from E into F satisfying(i) sup n ≥ k L n k < + ∞ and(ii) lim n →∞ L n ( g ) = T ( g ) for every g ∈ M ,it turns out that lim n →∞ L n ( f ) = T ( f ) for every f ∈ E. Thus, such subsets can be used to investigate the convergence of equibounded sequences ofpositive linear operators towards a given positive linear operator T : E −→ F or to approximateweakly T by means of (generally, simpler) linear operators L n , n ≥ Problem 5.3.
Given a positive linear operator T : E −→ F , find conditions under which thereexists a nontrivial (i.e., the linear subspace generated by it is not dense) Korovkin subset for T . Inthis case, try to determine some or all of them. orovkin-type theorems and positive operators Problem 5.4.
Given a subset M of E , try to determine some or all of the positive linear operators T : E → F (if they exist) for which M is a Korovkin subset.In the next sections, we shall discuss some aspects related to Problem 5.3 (for further details,we refer to [8, Sections 3.3 and 3.4]). As regards Problem 5.4, very few results are available (see,e.g., [71], [72-73], [74], [114-116]).The next result furnishes a complete characterization of Korovkin subsets for positive linearoperators in the setting of C ( X ) spaces. It was obtained by Yu. A. Shashkin ([110]) in thecase when X = Y , X compact metric space and T = I the identity operator, by H. Berens andG. G. Lorentz ([33]) when X = Y , X topological compact space, T = I , by H. Bauer and K. Donner([31]) when X = Y , X locally compact space, T = I , by C. A. Micchelli ([85]) and M. D. Rusk([105]) when X = Y , X compact, and by F. Altomare ([4]) in the general form below.We recall that M + b ( X ) denotes the cone of all bounded Radon measures on X (see Appendix). Theorem 5.5.
Let X and Y be locally compact Hausdorff spaces. Further, assume that X has acountable base and Y is metrizable. Given a positive linear operator T : C ( X ) −→ C ( Y ) and asubset M of C ( X ) , the following statements are equivalent:(i) M is a Korovkin subset of C ( X ) for T .(ii) If µ ∈ M + b ( X ) and y ∈ Y satisfy µ ( g ) = T ( g )( y ) for every g ∈ M , then µ ( f ) = T ( f )( y ) forevery f ∈ C ( X ) . Proof. (i) ⇒ (ii). Fix µ ∈ M + b ( X ) and y ∈ Y satisfying µ ( g ) = T ( g )( y ) for every g ∈ M . Considera decreasing countable base ( U n ) n ≥ of open neighborhoods of y in Y and, for every n ≥
1, choose ϕ n ∈ K ( Y ) such that: 0 ≤ ϕ n ≤ , ϕ n ( y ) = 1 and supp( ϕ n ) ⊂ U n (see Theorem 11.1 of theAppendix).Accordingly, define L n : C ( X ) −→ C ( Y ) by L n ( f ) := µ ( f ) ϕ n + T ( f )(1 − ϕ n ) ( f ∈ C ( X )) . Each L n is linear, positive and k L n k ≤ k µ k + k T k . Moreover, if g ∈ M , thenlim n →∞ L n ( g ) = T ( g ) in C ( Y ) , because, given ε >
0, there exists v ∈ N such that | T ( g )( z ) − T ( g )( y ) | ≤ ε for every z ∈ U v . Hence, since for every n ≥ v (thus U n ⊂ U v ) and for every z ∈ Y | L n ( g )( z ) − T ( g )( z ) | = ϕ n ( z ) | T ( g )( z ) − T ( g )( y ) | , we get | L n ( g )( z ) − T ( g )( z ) | = ( z U n , ≤ ε if z ∈ U n , and so k L n ( g ) − T ( g ) k ≤ ε . . Altomare M is a Korovkin subset for T , it turns out that, for every f ∈ C ( X ) , lim n →∞ L n ( f ) = T ( f )and hence lim n →∞ L n ( f )( y ) = T ( f )( y ). But, for every n ≥ L n ( f )( y ) = µ ( f ) and this completes theproof of (ii).(ii) ⇒ (i). Our proof starts with the observation that from statement (ii), it follows thatif µ ∈ M + b ( X ) and µ ( g ) = 0 for every g ∈ M, then µ = 0 . (5.1)Moreover, since X has a countable base, every bounded sequence in M + b ( X ) has a vaguely conver-gent subsequence (see Theorem 11.8 of Appendix). Consider now a sequence ( L n ) n ≥ of positivelinear operators from C ( X ) into C ( Y ) satisfying properties (i) and (ii) of Definition 5.2 andassume that for some f ∈ C ( X ) the sequence ( L n ( f )) n ≥ does not converge uniformly to T ( f ).Therefore, there exist ε >
0, a subsequence ( L k ( n ) ) n ≥ of ( L n ) n ≥ and a sequence ( y n ) n ≥ in Y such that | L k ( n ) ( f )( y n ) − T ( f )( y n ) | ≥ ε for every n ≥ . (5.2)We discuss separately the two cases when ( y n ) n ≥ is converging to the point at infinity of Y or not(see the Appendix).In the first case (which can only occur when Y is noncompact), we have lim n →∞ h ( y n ) = 0 forevery h ∈ C ( Y ).For every n ≥
1, define µ n ∈ M + b ( X ) by µ n ( f ) := L k ( n ) ( f )( y n ) ( f ∈ C ( X )) . Since k µ n k ≤ k L k ( n ) k ≤ M := sup n ≥ k L n k , replacing, if necessary, the sequence ( µ n ) n ≥ with asuitable subsequence, we may assume that there exists µ ∈ M + b ( X ) such that µ n −→ µ vaguely.But, if g ∈ M , then | µ n ( g ) | ≤ | L k ( n ) ( g )( y n ) − T ( g )( y n ) | + | T ( g )( y n ) | ≤≤ k L k ( n ) ( g ) − T ( g ) k + | T ( g )( y n ) | so that µ ( g ) = lim n →∞ µ n ( g ) = 0 . From (5.1) it turns out that µ ( f ) = 0 as well and hence | L k ( n ) ( f )( y n ) − T ( f )( y n ) | = | µ n ( f ) − T ( f )( y n ) | −→ y n ) n ≥ does not converge to the point at infinity of Y . Then, by replacing it with a suitable subsequence, we may assume that it converges to some y ∈ Y .Again, consider for every n ≥ µ n ( f ) := L k ( n ) ( f )( y n ) ( f ∈ C ( X )) . As in the previous reasoning, we may assume that there exists µ ∈ M + b ( X ) such that µ n −→ µ vaguely. Then, for g ∈ M , since | µ n ( g ) − T ( g )( y n ) | ≤ k L k ( n ) ( g ) − T ( g ) k −→ , orovkin-type theorems and positive operators µ ( g ) = T ( g )( y ). Therefore, assumption (ii) implies µ ( f ) = T ( f )( y ), or equivalently,lim n →∞ (cid:16) L k ( n ) ( f )( y n ) − T ( f )( y n ) (cid:17) = 0which is impossible because of (5.2). (cid:3) Some applications of Theorem 4.5 will be shown in subsequent sections. C ( X ) In this section, we discuss more closely those subsets of C ( X ) that are Korovkin subsets in C ( X )(see Definition 4.1), i.e., that are Korovkin subsets for the identity operator on C ( X ).Throughout the whole section, we shall fix a locally compact Hausdorff space with a countablebase, which is then metrizable as well. The next result immediately follows from Theorem 4.5. Theorem 6.1. ([31]). Given a subset M of C ( X ) , the following statements are equivalent:(i) M is a Korovkin subset of C ( X ) .(ii) If µ ∈ M + b ( X ) and x ∈ X satisfy µ ( g ) = g ( x ) for every g ∈ M , then µ ( f ) = f ( x ) for every f ∈ C ( X ) , i.e., µ = δ x (see (2.11)). In order to discuss a first application of Theorem 6.1, we recall that a mapping ϕ : Y −→ X between two locally compact Hausdorff spaces Y and X is said to be proper if for every compactsubset K ∈ X , the inverse image ϕ − ( K ) := { y ∈ Y | ϕ ( y ) ∈ K } is compact in Y . In this case, f ◦ ϕ ∈ C ( Y ) for every f ∈ C ( X ). Corollary 6.2.
Let Y be a metrizable locally compact Hausdorff space. If M is a Korovkin subsetof C ( X ) , then M is a Korovkin subset for any positive linear operator T : C ( X ) −→ C ( Y ) ofthe form T ( f ) := λ ( f ◦ ϕ ) ( f ∈ C ( X )) where λ ∈ C b ( Y ) , λ ≥ , and ϕ : Y −→ X is a proper mapping. Proof.
According to Theorem 5.5, we have to show that, if µ ∈ M + b ( X ) and y ∈ Y satisfy µ ( g ) = λ ( y ) g ( ϕ ( y )) for every g ∈ M , then µ ( f ) = λ ( y ) f ( ϕ ( y )) for every f ∈ C ( X ).If λ ( y ) = 0, then µ = 0 on M and hence µ = 0 by Theorem 6.1 and property (1) in the proof( ii ) = ⇒ ( i ) of Theorem 5.5. If λ ( y ) >
0, it suffices to apply Theorem 6.1 to 1 λ ( y ) µ and ϕ ( y ). (cid:3) Note that, if Y is a closed subset of X , then the canonical mapping ϕ : Y −→ X defined by ϕ ( y ) := y ( y ∈ Y ) is proper. Therefore, from Corollary 5.2, we get: Corollary 6.3.
Let M be a Korovkin subset of C ( X ) . Consider an equibounded sequence ( L n ) n ≥ of positive linear operators from C ( X ) into C ( X ) . Then the following properties hold.1) Given a closed subset Y of X and λ ∈ C b ( Y ) , λ ≥ , if lim n →∞ L n ( g ) = λg uniformly on Y forevery g ∈ M , then lim n →∞ L n ( f ) = λf uniformly on Y for every f ∈ C ( X ) . . Altomare
2) If lim n →∞ L n ( g ) = g uniformly on compact subsets of X for every g ∈ M , then lim n →∞ L n ( f ) = f uniformly on compact subsets of X for every f ∈ C ( X ) . Next, we proceed to investigate some useful criteria to explicitly determine Korovkin subsetsof C ( X ). We begin with the following result which is at the root of all subsequent results. Werecall that, if M is a subset of C ( M ), then the symbol L ( M ) denotes the linear subspace of C ( X )generated by M . Proposition 6.4.
Let M be a subset of C ( X ) and assume that for every x, y ∈ X, x = y , thereexists h ∈ L ( M ) , h ≥ , such that h ( x ) = 0 and h ( y ) > . Then M is a Korovkin subset of C ( X ) . Proof.
We shall verify condition (ii) of Theorem 6.1. Therefore, consider µ ∈ M + b ( X ) and x ∈ X satisfying µ ( g ) = g ( x ) for every g ∈ M and, hence, for every g ∈ L ( M ).If y ∈ X, y = x , then there exists h ∈ L ( M ), h ≥
0, such that h ( y ) > h ( x ) = µ ( h ).Therefore, by Theorem 11.7 of the Appendix, there exists α ≥ µ = αδ x . Choosing h ∈ L ( M ) such that h ( x ) >
0, we get αh ( x ) = µ ( h ) = h ( x ), so α = 1 and µ = δ x . (cid:3) Below, we state an important consequence of Proposition 6.4. In the sequel, if M is a subset of C ( X ) and for f ∈ C ( X ), we shall set f M := { f · f | f ∈ M } (6.1)and f M := { f · f | f ∈ M } . (6.2) Theorem 6.5.
Consider a strictly positive function f ∈ C ( X ) and a subset M of C ( X ) thatseparates the points of X . Furthermore, assume that f M ∪ f M ⊂ C ( X ) . Then { f } ∪ f M ∪ f M is a Korovkin subset of C ( X ) .If, in addition, M is finite, say M = { f , . . . , f n } , n ≥ , then ( f , f f , . . . , f f n , f n X i =1 f i ) is a Korovkin subset of C ( X ) . Proof. If x, y ∈ X, x = y , then there exists f ∈ M such that f ( x ) = f ( y ). Therefore, the function h := f ( f − f ( x )) belongs to L ( { f } ∪ f M ∪ f M ), it is positive and h ( x ) = 0 < h ( y ). Thus theresult follows from Proposition 6.4.If M = { f , . . . , f n } is finite, then in the above reasoning one can consider the function h := f n P i =1 ( f i − f i ( x )) . (cid:3) The next result is an obvious consequence of Theorem 6.5 but is worth being stated explicitlybecause of its connection with the Stone-Weierstrass theorem (see Section 9).
Theorem 6.6.
Consider a strictly positive function f ∈ C ( X ) and a subset M of C ( X ) thatseparates the points of X . Then { f } ∪ f M ∪ f M is a Korovkin subset of C ( X ) .Moreover, if M is finite, say M = { f , . . . , f n } , n ≥ , then ( f , f f , . . . , f f n , f n X i =1 f i ) orovkin-type theorems and positive operators is a Korovkin subset of C ( X ) .Finally, if f is also injective, then { f , f , f } is a Korovkin subset of C ( X ) . From Theorem 6.6, the following result immediately follows.
Corollary 6.7. (1) { e , e , e } is a Korovkin subset of C (]0 , . (2) { e − , e − , e − } is a Korovkin subset of C ([1 , + ∞ [) , where e − k ( x ) := x − k for every x ∈ [1 , + ∞ [ and k = 1 , , .(3) { f , f , f } is a Korovkin subset of C ([0 , + ∞ [) where f k ( x ) := exp ( − kx ) for every x ∈ [0 , + ∞ [ and k = 1 , , .(4) { Φ , pr Φ , . . . , pr d Φ , k · k Φ } is a Korovkin subset of C ( R d ) , d ≥ , where Φ( x ) := exp ( −k x k ) for every x ∈ R d . A useful generalization of the previous result is presented below.
Proposition 6.8.
Given λ , λ , λ ∈ R , < λ < λ < λ , then(1) { e λ , e λ , e λ } is a Korovkin subset of C (]0 , where e λ k ( x ) := x λ k for every x ∈ ]0 , and k = 1 , , .(2) { e − λ , e − λ , e − λ } is a Korovkin subset of C ([1 , + ∞ [) where e − λ k ( x ) := x − λ k for every x ∈ [1 , + ∞ [ and k = 1 , , .(3) { f λ , f λ , f λ } is a Korovkin subset of C ([0 , + ∞ [) where f λ k ( x ) := exp( − λ k x ) for every x ∈ [0 , + ∞ [ and k = 1 , , . Proof.
We give the proof only for (3); the proofs of the other statements are left to the reader(see, e.g., [8, Proposition 4.2.4]). We shall apply Proposition 6.4 and to this end fix x ∈ [0 , + ∞ [.Then, by using differential calculus, it is not difficult to show that the function h ( x ) := exp( − λ x ) + α exp( − λ x ) + β exp( − λ x ) ( x ≥ α := λ − λ λ − λ exp(( λ − λ ) x ) and β := λ − λ λ − λ exp(( λ − λ ) x ), satisfies h ( x ) = 0 < h ( y )for every y ∈ [0 , + ∞ [, y = x . (cid:3) Here, we discuss some applications of Proposition 6.8. The first application is taken from [31,Proposition 4.1].We begin by recalling that, if ϕ ∈ L ([0 , + ∞ [), then the Laplace transform of ϕ on [0 , + ∞ [is defined by L ( ϕ )( λ ) := Z + ∞ e − λt ϕ ( t ) d t ( λ ≥ . (6.3)By Lebesgue’s dominated convergence theorem, L ( ϕ ) ∈ C ([0 , + ∞ [) and kL ( ϕ ) k ≤ k ϕ k . By meansof ϕ , we may naturally define a linear operator L ϕ : C ([0 , + ∞ [) → C ([0 , + ∞ [) by setting for every f ∈ C ([0 , + ∞ [) and x ≥ L ϕ ( f )( x ) := Z + ∞ f ( x + y ) ϕ ( y ) d y = Z + ∞ x f ( u ) ϕ ( x − u ) d u. (6.4) . Altomare L ϕ ( f ) ∈ C ([0 , + ∞ [) by virtue of Lebesgue’s dominated convergence theorem. Moreover L ϕ is bounded and k L ϕ k ≤ k ϕ k . Finally, if ϕ ≥
0, then L ϕ is positive as well.For every λ >
0, denoting by f λ the function f λ ( x ) := exp( − λx ) ( x ≥ , (6.5)we get L ϕ ( f λ ) = L ( ϕ )( λ ) f λ . (6.6) Proposition 6.9.
Consider a sequence ( ϕ n ) n ≥ of positive functions in L ([0 , + ∞ [) such that sup n ≥ k ϕ n k < + ∞ and, for every n ≥ , denote by L n : C ([0 , + ∞ [) → C ([0 , + ∞ [) the positive linear operator associated with ϕ n defined by (6.4). Then the following statements areequivalent:(i) lim n →∞ L n ( f ) = f uniformly on [0 , + ∞ [ for every f ∈ C ([0 , + ∞ [) .(ii) lim n →∞ L ( ϕ n )( λ ) = 1 for every λ > .(iii) There exist λ , λ , λ ∈ R , < λ < λ < λ such that lim n →∞ L ( ϕ n )( λ k ) = 1 for k = 1 , , . Proof.
The implication (i) ⇒ (ii) follows from (6.6). The implication (ii) ⇒ (iii) being obvious, theonly point remaining concerns (iii) ⇒ (i).Since ( ϕ n ) n ≥ is bounded in L ([0 , + ∞ [), we have sup n ≥ k L n k < + ∞ . Moreover, (iii) means thatlim n →∞ L n ( f λ k ) = f λ k uniformly on [0 , + ∞ [) by (6.6). Therefore, (i) follows by applying part (3) ofProposition 6.8. (cid:3) As another application of the previous results, we shall study the behaviour of the Sz´asz-Mirakjan operators (see (4.43)) on C ([0 , + ∞ [) and on continuous function spaces on [0 , + ∞ [ withpolynomial weights.We recall that these operators are defined by M n ( f )( x ) := exp( − nx ) ∞ X k =0 f (cid:16) kn (cid:17) n k x k k ! (6.7)for n ≥ , x ≥ f ∈ C ([0 , + ∞ [) such that | f ( x ) | ≤ M exp( αx ) ( x ≥
0) for some M ≥ α > Lemma 6.10. If f ∈ C ([0 , + ∞ [) , then M n ( f ) ∈ C ([0 , + ∞ [) and k M n ( f ) k ≤ k f k for every n ≥ . Proof.
The function M n ( f ) is continuous because the series (6.7) is uniformly convergent oncompact subsets of [0 , + ∞ [. Moreover, for every x ≥ | M n ( f )( x ) | ≤ k f k exp( − nx ) ∞ X k =0 n k x k k ! = k f k . orovkin-type theorems and positive operators M n ( f ) ∈ C ([0 , + ∞ [), given ε >
0, choose v ∈ N such that | f (cid:16) kn (cid:17) | ≤ ε forevery k ≥ v . For sufficiently large x ≥ − nx ) v X k =0 (cid:12)(cid:12)(cid:12) f (cid:16) kn (cid:17)(cid:12)(cid:12)(cid:12) ( nx ) k k ! ≤ ε so that | M n ( f )( x ) (cid:12)(cid:12)(cid:12) ≤ exp( − nx ) v X k =0 (cid:12)(cid:12)(cid:12) f (cid:16) kn (cid:17)(cid:12)(cid:12)(cid:12) ( nx ) k k ! + exp( − nx ) ∞ X k = v +1 (cid:12)(cid:12)(cid:12) f (cid:16) kn (cid:17)(cid:12)(cid:12)(cid:12) ( nx ) k k ! ≤ ε. (cid:3) Theorem 6.11.
For every f ∈ C ([0 , + ∞ [) , lim n →∞ M n ( f ) = f uniformly on [0 , + ∞ [ . Proof.
Since ( M n ) n ≥ is an equibounded sequence of positive linear operators from C ([0 , + ∞ [)into C ([0 , + ∞ [), by Proposition 6.8, (3), it is sufficient to show that, for every λ ≥ n →∞ M n ( f λ ) = f λ uniformly on [0 , + ∞ [where f λ is defined by (6.5).A simple calculation indeed shows that M n ( f λ )( x ) = exp h − λx (cid:16) − exp( − λ/n ) λ/n (cid:17)i for every x ≥
0. Since lim n →∞ − exp( − λ/n ) λ/n = 1, the sequence ( M n ( f λ )) n ≥ converges pointwise to f λ and is decreasing. Moreover, each M n ( f λ ) and f λ vanishes at + ∞ and so, by Dini’s theoremapplied in the framework of the compactification [0 , + ∞ ], we obtain the uniform convergence aswell. (cid:3) Our next aim is to discuss the behaviour of the operators M n on continuous function spaceswith polynomial weights. This naturally leads to investigate some Korovkin-type results in weightedcontinuous function spaces. For further results in this respect, we also refer to the next Section 8and to [10-11], [14], [60-62], [104].Consider again an arbitrary locally compact Hausdorff space X having a countable base. Let w be a continuous weight on X , i.e., w ∈ C ( X ) and w ( x ) > x ∈ X , and set C w ( X ) := { f ∈ C ( X ) | wf ∈ C ( X ) } . (6.8)The space C w ( X ), endowed with the natural (pointwise) order and the weighted norm k f k w := k wf k ∞ ( f ∈ C w ( X )) (6.9)is a Banach lattice . Altomare w ∈ C b ( X ), then C ( X ) ⊂ C w ( X ) and, if w ∈ C ( X ), then C b ( X ) ⊂ C w ( X ). The spaces C w ( X ) and C ( X ) are lattice isomorphic, a lattice isomorphism between C ( X ) and C w ( X ) beingthe linear operator S : C ( X ) → C w ( X ) defined by S ( f ) := w − f ( f ∈ C ( X )) . (6.10)Therefore, if M is a subset of C w ( X ) , then M is a Korovkin subset of C w ( X ) if and only if wM := { wf | f ∈ M } (6.11) is a Korovkin subset of C ( X ).Accordingly, from Proposition 6.4 and Theorem 6.5, we immediately obtain the following result. Corollary 6.12.
Given a subset M of C w ( X ) , the following statements hold:(1) If for every x, y ∈ X , x = y , there exists h ∈ L ( M ) , h ≥ , such that h ( x ) = 0 < h ( y ) , then M is a Korovkin subset of C w ( X ) .(2) Assume that w ∈ C ( X ) . If M ⊂ C w ( X ) and M separates the points of X , then { } ∪ M ∪ M is a Korovkin subset of C w ( X ) . Moreover, if M = { f , . . . , f n } , n ≥ , is finite, then (cid:26) , f , . . . , f n , n P i =1 f i (cid:27) is a Korovkin subset of C w ( X ) . Applying the above part (2) to a subset X of R d , d ≥
1, and M = { pr , . . . , pr d } , we obtain: Corollary 6.13.
Consider a locally compact subset X of R d , d ≥ , and w ∈ C ( X ) such that k · k w ∈ C ( X ) . Then { , pr , . . . , pr d , k · k } is a Korovkin subset of C w ( X ) . From the previous Corollary 6.13, it follows in particular that, if X is a noncompact real interval,then { , e , e } is a Korovkin subset of C w ( X ) for any w ∈ C ( X ) such that e w ∈ C ( X ) . When X ⊂ [0 , + ∞ [, this result can be considerably generalized. Corollary 6.14.
Let X be a noncompact subinterval of [0 , + ∞ [ and let w ∈ C ( X ) be a weight on X . Consider λ , λ ∈ R such that < λ < λ and e λ ∈ C w ( X ) . Then { , e λ , e λ } is a Korovkinsubset of C w ( X ) . Proof.
First note that e λ ∈ C w ( X ) because x λ ≤ x λ for every x ≥
0. In order to get theresult, we shall apply part (1) of Corollary 6.12 by showing that for a given x ∈ X there exists h ∈ L ( { , e λ , e λ ) such that h ( x ) = 0 < h ( y ) for any y ∈ X, y = x .Set a := inf X ≥
0. If x = a , then it is sufficient to consider h ( x ) = x λ − a λ ( x ∈ X ). If X is upper bounded and x = sup X , then we may consider h ( x ) = x λ − x λ ( x ∈ X ). Finally if x belongs to the interior of X , then by means of differential calculus it is not difficult to show thatthe function h ( x ) := ( λ − λ ) x λ − λ x λ − λ x λ + λ x λ ( x ∈ X )satisfies the required properties. (cid:3) From (6.11) and Proposition (6.8), (3), we also get: orovkin-type theorems and positive operators
Corollary 6.15.
Consider the functions g β ( x ) := exp( βx ) and g γ ( x ) := exp( γx ) ( x ∈ [0 , + ∞ [) , where < β < γ. Then for every α > γ, the subset { , g β , g γ } is a Korovkin subset of the space E α := { f ∈ C ([0 , + ∞ [) | lim x → + ∞ exp( − αx ) f ( x ) = 0 } endowed with the weighted norm k f k α := sup x ≥ exp( − αx ) | f ( x ) | ( f ∈ E α ) ). In the case when w ∈ C b ( X ), another easy method of finding Korovkin subsets of C w ( X ) isindicated below. Proposition 6.16.
Consider a continuous bounded weight w ∈ C b ( X ) . Then every Korovkinsubset of C ( X ) is a Korovkin subset of C w ( X ) as well. Proof.
Let M be a Korovkin subset of C ( X ). In order to show that M is a Korovkin subsetof C w ( X ), we shall show that wM is a Korovkin subset of C ( X ). To this end, fix µ ∈ M + ( X )and x ∈ X such that µ ( wg ) = w ( x ) g ( x ) for every g ∈ M , and consider the positive linear form ν : C ( X ) −→ R defined by ν ( f ) := µ ( wf ) /w ( f ∈ C ( X )). Then ν ∈ M + b ( X ) and ν ( g ) = g ( x )for every g ∈ M . From Theorem 6.1, we then conclude that ν ( f ) = f ( x ) for every f ∈ C ( X ), i.e., µ ( wf ) = w ( x ) f ( x ) for every f ∈ C ( X ).Note that if ϕ ∈ K ( X ), after setting f := ϕ/w ∈ K ( X ), we get ϕ = wf and hence µ ( ϕ ) = ϕ ( x ).By density, we conclude that µ ( f ) = f ( x ) for every f ∈ C ( X ) and hence the desired result followsfrom Theorem 6.1. (cid:3) Two simple applications of Corollary 6.13 and Proposition 6.16 are shown below. The first oneis concerned again with the Sz´asz-Mirakjan operators (6.7) on the weighted continuous functionspaces C w m ([0 , + ∞ [) := (cid:26) f ∈ C ([0 , + ∞ [) | lim x →∞ f ( x )1 + x m = 0 (cid:27) , (6.12) m ≥
1, where w m ( x ) := x m ( x ≥ . Theorem 6.17.
For every m ≥ and n ≥ and for every f ∈ C w m ([0 , + ∞ [) , M n ( f ) ∈ C w m ([0 , + ∞ [) and lim n →∞ M n ( f ) = f on [0 , + ∞ [ with respect to the weighted norm k · k w m and,hence, uniformly on compact subsets of [0 , + ∞ [ . Proof.
In [32, Lemma 5], it was shown that every M n is a bounded linear operator from C w m into C w m and that sup x ≥ k M n k < + ∞ . For every λ > f λ ( x ) :=exp( − λx ) ( x ≥ { f λ , f λ , f λ } is a Korovkin subsetof C w m ([0 , + ∞ [) provided that 0 < λ < λ < λ . On the other hand, for each function f λ , wehave already shown that lim n →∞ M n ( f λ ) = f λ uniformly on [0 , + ∞ [ (see the proof of Theorem 6.11)and hence the same limit relationship holds with respect to k · k w m because k · k w m ≤ k · k ∞ on C ([0 , + ∞ [) and hence the result follows. (cid:3) Remarks 6.18.
1. The behaviour of Sz´asz-Mirakjan operators on locally convex weighted function spaces hasbeen further investigated in [11]. . Altomare B n ( f )( x ) := 1(1 + x ) n n X h =0 (cid:18) n + h − h (cid:19) f (cid:16) hn (cid:17)(cid:16) x x (cid:17) h (6.13)( f ∈ C w m ([0 , + ∞ [), x ≥ m ≥ n ≥ X = R and the weight w m ( x ) := x m ( x ∈ R ) with m ≥ m even. Consider the sequence of the Gauss-Weierstrass operators defined by (4.44) for p = 1. Hence G n ( f )( x ) := (cid:16) n π (cid:17) / Z R f ( t ) exp (cid:16) − n t − x ) (cid:17) d t (6.14)for n ≥ x ∈ R where f : R −→ R is any Borel measurable function for which the integral(6.14) is absolutely convergent. In particular, the operators G n are well-defined on the functionspace C w m ( R ) := (cid:26) f ∈ C ( R ) | lim | x |→∞ f ( x )1 + x m = 0 (cid:27) . (6.15) Theorem 6.19.
Under the above hypotheses, G n ( C w n ( R )) ⊂ C w m ( R ) and lim n →∞ G n ( f ) = f with respect to k · k w m (and hence uniformly on compact subsets of R ) for any f ∈ C w m ( R ) . Proof.
Consider the function e m ( x ) = x m ( x ∈ R ). Then, for any n ≥ x ∈ R , G n ( e m )( x ) = (cid:16) n π (cid:17) / Z R t m exp (cid:16) − n t − x ) (cid:17) d t = m X k =0 (cid:18) mk (cid:19) M k (cid:16) n (cid:17) k/ x m − k where M := 1 and M k := ( k is odd , · · · · ( k −
1) if k is even , (see [29, formulae (4.20) and (4.21)]). Therefore, G n ( e m )( x ) ≤ m X k =0 (cid:18) mk (cid:19) M k (cid:16) n (cid:17) k/ | x | m − k ≤ x m + m X k =1 (cid:18) mk (cid:19) M k k/ | x | m − k and w m ( x ) G n (1 + e m )( x ) ≤
11 + x m (cid:16) x m + m X k =1 (cid:18) mk (cid:19) M k k/ | x | m − k (cid:17) ≤ M,M being a suitable positive constant independent of m .Consider now f ∈ C w m ( R ). Thus to each ε > δ > x ∈ R , | x | ≥ δ , w m ( x ) | f ( x ) | ≤ ε M . orovkin-type theorems and positive operators δ ≥ δ such that w m ( x ) ≤ ε M δ for x ∈ R , | x | ≥ δ, where M δ := sup | x |≤ δ | f ( t ) | . Then, for any x ∈ R , | x | ≥ δ , | G n ( f )( x ) | ≤ (cid:16) n π (cid:17) / Z | t |≤ δ | f ( t ) | exp (cid:16) − n t − x ) (cid:17) d t + (cid:16) n π (cid:17) / Z | t |≥ δ | f ( t ) | exp (cid:16) − n t − x ) (cid:17) d t ≤ M δ + ε M (cid:16) n π (cid:17) / Z | t |≥ δ (1 + e m ( t )) exp (cid:16) − n t − x ) (cid:17) d t ≤ M δ + ε M G n (1 + e m )( x )so that w m ( x ) | G n ( f )( x ) | ≤ ε because of the previous estimates.This proves that G n ( f ) ∈ C w m ( R ). Moreover, note that, since | f | ≤ k f k w m (1 + e m ), then w m | G n ( f ) | ≤ k f k w m w m G n (1 + e m )and hence k G n ( f ) k w m ≤ M k f k w m , that is, k G n k ≤ M for any n ≥ { , e , e } is a Korovkin subset of C w m ( R ),where e k ( x ) := x k ( k = 1 , x ∈ R ), and for every h ∈ { , e , e } clearly G n ( h ) → h as n → ∞ with respect to k · k w n because of (4.48)-(4.50). Therefore, the result follows. (cid:3) Remark 6.20.
A generalization of Theorem 6.19 can be found in [21, Theorem 3.1] and [23]. C ( X ) , X compact The results of the previous section also apply when X is a compact metric space (and hence C ( X ) = C ( X )). However, in this particular case, some of them have a particular relevance andhence are worth an explicit description.From Theorem 6.5, by replacing f with the constant function , the following result immedi-ately follows. Theorem 7.1. If M is a subset of C ( X ) that separates the points of X , then { } ∪ M ∪ M is aKorovkin subset of C ( X ) .Moreover, if M = { f , . . . , f n } , n ≥ , is finite, then (cid:26) , f , . . . , f n , n P i =1 f i (cid:27) is a Korovkin subsetof C ( X ) .In particular, if f ∈ C ( X ) is injective, then { , f, f } is a Korovkin subset of C ( X ) . Theorem 7.1 extends both the Korovkin Theorems 3.1 and 4.3 as well as Volkov’s Theorem 4.2.It was obtained by Freud ([59]) for finite subsets M and by Schempp ([106]) in the general case.By adapting the proof of Proposition 6.8 and by using Proposition 6.4, it is not difficult to showthat: . Altomare Proposition 7.2. (1) If < λ < λ , then { , e λ , e λ } is a Korovkin subset of C ([0 , , where e λ k ( x ) := x λ k for every x ∈ [0 , and k = 1 , .(2) If u ∈ C ([0 , is strictly convex, then { , e , u } is a Korovkin subset of C ([0 , . Next, we shall discuss some applications of Theorem 7.1 by considering a convex compact subset K of a locally convex space E . We denote by A ( K )the subspace of all real-valued continuous affine functions on K . We recall that a function u : K → R is said to be affine if u ( λx + (1 − λ ) y ) = λu ( x ) + (1 − λ ) u ( y ) (7.1)for every x, y ∈ K and λ ∈ [0 , , or, equivalently, if u ( n X i =1 λ i x i ) = n X i =1 λ i u ( x i ) (7.2)for every n ≥ x , . . . , x n ∈ K and λ , . . . , λ n ≥ P ni =1 λ i = 1 . Note that A ( K ) contains the constant functions as well as the restrictions to K of every con-tinuous linear functional on E so that, by the Hahn-Banach theorem, A ( K ) separates the pointsof K . Therefore, from Theorem 7.1, we immediately get: Corollary 7.3. A ( K ) ∪ A ( K ) is a Korovkin subset of C ( K ) . Consider now a continuous selection ( e µ x ) x ∈ K of probability Borel measures on K , i.e., for every f ∈ C ( K ) the function x R K f d e µ x is continuous on K . Such a function will be denoted by T ( f ),that is T ( f )( x ) := Z K f d e µ x ( x ∈ K ) . (7.3)Note that T can be viewed as a positive linear operator from C ( K ) into C ( K ) and T ( ) = .Conversely, by the Riesz representation theorem (see Theorem 11.3 of the Appendix), eachpositive linear operator T : C ( K ) −→ C ( K ) generates a continuous selection of probability Borelmeasures on K satisfying (7.3). Such a selection will be referred to as the canonical continuousselection associated with T .From now on, we fix a continuous selection ( e µ x ) x ∈ K of probability Borel measures on K satis-fying Z K u d e µ x = u ( x ) for every u ∈ A ( K ) , (7.4)namely, T ( u ) = u for every u ∈ A ( K ) . (7.5)Property (7.4) means that each x ∈ K is the barycenter or the resultant of e µ x ([39, Vol. II,Def. 26.2]).For every n ≥ x ∈ K , denote by e µ ( n ) x orovkin-type theorems and positive operators K n of e µ x with itself n times (see, e.g., [30, § f ∈ C ( K ), B n ( f )( x ) := Z K n f (cid:16) x + · · · + x n n (cid:17) d e µ ( n ) x ( x , . . . , x n ) . (7.6)By Fubini’s theorem, we can also write B n ( f )( x ) = Z K · · · Z K f (cid:16) x + · · · + x n n (cid:17) d e µ x ( x ) · · · d e µ x ( x n ) . (7.7)By the continuity property of the product measure (see [39, Vol. I, Proposition 13.12] and [30,Theorem 30.8]) and the continuity of the selection, it follows that B n ( f ) ∈ C ( K ).The positive linear operator B n : C ( K ) −→ C ( K ) is referred to as the n -th Bernstein-Schnabl operator associated with the given selection ( e µ x ) x ∈ K (or the given positive linear oper-ator T : C ( K ) −→ C ( K )) . Bernstein-Schnabl operators were introduced for the first time in 1968 by Schnabl ([107-108])in the context of the set of all probability Radon measures on a compact Hausdorff space and, aswe shall see next, they generalize the classical Bernstein operators (3.5). Subsequently, Grossman([66]) introduced the general definition (7.6) (or (7.7)). In the particular case of Bauer simplices(see, e.g., [8, Section 1.5, p. 59]), these operators have been extensively studied by Nishishiraho ([87-91]). Another particular class of them has been also studied by Altomare ([5]). Their constructionessentially involves positive projections and, in this case, they satisfy many additional propertiesuseful in the study of evolution problems (see the end of Section 10). For some shape preservingproperties of these operators, we also refer to Ra¸sa ([97-99]).For a comprehensive survey on these operators, we refer to [8, Chapter 6] and to the referencescontained in the relevant notes. More recent results can be also found in [7], [18], [19], [24-25],[100-102], [103].Below, we discuss some examples.
Examples 7.4.
1. Consider K = [0 ,
1] and set e µ x := xδ + (1 − x ) δ for all x ∈ [0 , e µ x ) ≤ x ≤ is a con-tinuous selection and the corresponding Bernstein-Schnabl operators turn into the Bernsteinoperators (3.5) (use an induction argument).2. Let α, β, γ ∈ C ([0 , ≤ α ≤ , ≤ β ≤ , ≤ γ ≤ , α + β + γ = and β ( x ) / γ ( x ) = x for every x ∈ [0 , e µ x := α ( x ) δ + β ( x ) δ / + γ ( x ) δ . Then theBernstein-Schnabl operators associated with ( e µ x ) ≤ x ≤ on [0 ,
1] are given by B n ( f )( x ) = n X h =0 n X k =0 (cid:18) nh (cid:19)(cid:18) n − hk (cid:19) α ( x ) n − h − k β ( x ) h γ ( x ) k f (cid:16) h + 2 k n (cid:17) (7.8)( f ∈ C ([0 , , ≤ x ≤ a = a < a < · · · < a p = b , p ≥
1, be a subdivision of the compact real interval [ a, b ].For every x ∈ [ a, b ], set e µ x := x − a k a k +1 − a k δ a k +1 + a k +1 − xa k +1 − a k δ a k , (7.9) . Altomare x ∈ [ a k , a k +1 ], 0 ≤ k ≤ p −
1. Then the Bernstein-Schnabl operators are given by B n ( f )( x ) = 1( a k +1 − a k ) n n X r =0 (cid:18) nr (cid:19) ( x − a k ) r ( a k +1 − x ) n − r f (cid:16) rn a k +1 + n − rn a k (cid:17) (7.10)whenever x ∈ [ a k , a k +1 ] , ≤ k ≤ p − f ∈ C ([ a, b ]) , n ≥ .
4. Consider the d -dimensional simplex K d of R d defined by (4.38) and for every x = ( x , . . . , x d ) ∈ K d , set e µ x := (cid:16) − d X i =1 x i (cid:17) δ + d X i =1 x i δ a i where a i = ( δ ij ) ≤ j ≤ d for every i = 1 , . . . , d, δ ij being the Kronecker symbol. Then theBernstein-Schnabl operators associated with ( e µ x ) x ∈ K d turn into the Bernstein operators onthe simplex K d defined by (4.38).5. Consider the hypercube Q d := [0 , d of R d and for every x = ( x , . . . , x d ) ∈ Q d , set e µ x := X h ,...,h d =0 x h (1 − x ) − h · · · x h d d (1 − x d ) − h d δ b h ,...,hd , where b h ,...,h d = ( δ h , . . . , δ h d ) ( h , . . . , h d ∈ { , } ) . Then the Bernstein-Schnabl operatorsare the Bernstein operators on Q d defined by (4.41).Many other significant examples can be described in the framework of other finite dimensionalsubsets such as balls, or more generally, ellipsoids of R d , or in the setting of infinite-dimensionalBauer simplices (see [8, Chapter 6], [24], [103]).In order to discuss the approximation properties of Bernstein-Schnabl operators, note that forevery n ≥ B n ( ) = . (7.11)Moreover, if u ∈ A ( K ), then for every x , . . . , x n ∈ Ku (cid:16) x + · · · + x n n (cid:17) = u ( x ) + · · · + u ( x n ) n and u (cid:16) x + · · · + x n n (cid:17) = 1 n h n X i =1 u ( x i ) + 2 X ≤ i For every f ∈ C ( K ) , lim n →∞ B n ( f ) = f uniformly on K. For additional properties of Bernstein-Schnabl operators, including estimates of the rate ofconvergence, asymptotic formulae, shape-preserving properties and, especially, their connectionswith the approximation of positive semigroups as well as of the solutions of evolution equations,we refer to the references we cited before Examples 7.4. L p ( X, e µ ) spaces By using Korovkin-type theorems in spaces of continuous functions it is also possible to get someresults for L p ( X, e µ )-spaces, 1 ≤ p < + ∞ . In this regard, other than the space C ( X ), also weightedcontinuous function spaces can be efficiently used. In this section, we shall develop this methodby also showing some additional Korovkin-type results in weighted continuous function spacesthat complement the ones already discussed at the end of Section 6, and then by presenting thecorresponding ones in L p ( X, e µ )-spaces.The method relies on a fundamental characterization of Korovkin subspaces which is due toBauer and Donner ([31]) (see also [8, Theorem 4.1.2]). Its counterpart for Korovkin subspacesfor positive linear operators was obtained in [8, Theorem 3.1.4]. An extension to locally convexfunction spaces was obtained in [10]. For the sake of brevity, we omit its proof.As in the previous section, X denotes a fixed locally compact Hausdorff space with a countablebase. Theorem 8.1. ([31]) Given a linear subspace H of C ( X ) , the following statements are equivalent:(i) H is a Korovkin subspace of C ( X ) ;(ii) For every f ∈ C ( X ) and for every ε > , there exist finitely many functions h , . . . , h n ∈ H , k , . . . , k n ∈ H and u, v ∈ C ( X ) , u, v ≥ such that k u k ≤ ε, k v k ≤ ε and (cid:13)(cid:13)(cid:13)(cid:13) inf ≤ j ≤ n k j − sup ≤ i ≤ n h i (cid:13)(cid:13)(cid:13)(cid:13) ≤ ε and sup ≤ i ≤ n h i − u ≤ f ≤ inf ≤ j ≤ n k j + v. Remark 8.2. If X is compact and ∈ H , then Theorem 8.1 can be stated in a simpler form (see[8, Theorem 4.1.4]).Let us mention two important consequences of Theorem 8.1. Theorem 8.3. ([31]). Let H be a Korovkin subspace of C ( X ) . If E is a Banach lattice and if S : C ( X ) → E is a lattice homomorphism, then H is a Korovkin subspace of C ( X ) for S . Proof. Consider a sequence ( L n ) n ≥ of positive linear operators from C ( X ) into E such that M := sup n ≥ k L n k < + ∞ and lim n →∞ L n ( h ) = S ( h ) for every h ∈ H . Given f ∈ C ( X ) and ε > δ ∈ ]0 , ε [ such that k S ( g ) k ≤ ε for every g ∈ C ( X ), k g k ≤ δ . By Theorem 8.1, there exist h , . . . , h p , k , . . . , k p ∈ H and u, v ∈ C ( X ), u, v ≥ 0, such that k u k ≤ δ , k v k ≤ δ and k v k ≤ ε . Altomare (cid:13)(cid:13)(cid:13)(cid:13) inf ≤ j ≤ n S ( k j ) − sup ≤ i ≤ n s ( h i ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ δ and sup ≤ i ≤ p h i − u ≤ f ≤ inf ≤ j ≤ p k j + v. Therefore, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) inf ≤ j ≤ p S ( k j ) − sup ≤ i ≤ p S ( h i ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε. Moreover, for every n ≥ ≤ i ≤ p L n ( h i ) − L n ( u ) ≤ L n ( f ) ≤ inf ≤ j ≤ p L n ( k j ) + L n ( v )and sup ≤ i ≤ p S ( h i ) − S ( u ) ≤ S ( f ) ≤ inf ≤ j ≤ p S ( k j ) + S ( v ) . Accordingly, L n ( f ) − S ( f ) ≤ p X j =0 | L n ( k j ) − S ( k j ) | + | inf ≤ j ≤ p S ( k j ) − sup ≤ i ≤ p S ( h i ) | + L n ( v ) + S ( u )and S ( f ) − L n ( f ) ≤ p X i =0 | L n ( h i ) − S ( h i ) | + | inf ≤ j ≤ p S ( k j ) − sup ≤ i ≤ p S ( h i ) | + + L n ( u ) + S ( v ) , so that | S ( f ) − L n ( f ) | ≤ p X i =0 | L n ( h i ) − S ( h i ) | + p X j =0 | L n ( k j ) − S ( k j ) | + | inf ≤ j ≤ p S ( k j ) − sup ≤ i ≤ p S ( h i ) | + L n ( u ) + L n ( v ) + S ( u ) + S ( v ) ≤ p X i =0 | L n ( h i ) − S ( h i ) | + p X j =0 | L n ( k j ) − S ( k j ) | + 2 M ε + 2 ε. It is now easy to conclude that L n ( f ) → S ( f ) as n → ∞ because L n ( h i ) → S ( h i ) and L n ( k j ) → S ( k j ) as n → ∞ for every i, j = 0 , . . . , p . (cid:3) Remark 8.4. From Theorem 8.3, it follows that, if M is a Korovkin subset of C ( X ), then M isa Korovkin subset for the natural embedding from C ( X ) into an arbitrary closed lattice subspace E of B ( X ) containing C ( X ) (for instance, E = C b ( X ) or E = B ( X )).Another consequence of Theorem 8.3 is indicated below. Proposition 8.5. ([120]). Let E be a Banach lattice and consider a lattice homomorphism S : C ( X ) → E such that S ( C ( X )) is dense in E . If M is a Korovkin subspace of C ( X ) , then S ( M ) is a Korovkin subspace of E . orovkin-type theorems and positive operators Proof. Consider an equibounded sequence ( L n ) n ≥ of positive linear operators on E and assumethat L n ( k ) → k as n → ∞ for every k ∈ S ( M ). This means that L n ( S ( h )) → S ( h ) as n → ∞ foreach h ∈ M and hence for each h ∈ H := L ( M ). So, by Theorem 8.3, L n ( S ( f )) −→ S ( f ) for every f ∈ C ( X ). The result now follows from the assumption that S ( C ( X )) is dense in E and ( L n ) n ≥ is equibounded. (cid:3) Remark 8.6. A typical situation where Proposition 8.5 can be applied concerns the case when E is a Banach lattice containing C ( X ) as a dense sublattice and S is the natural embedding from C ( X ) into E . Thus, in this case, every Korovkin subspace of C ( X ) is a Korovkin subspace of E as well .After these preliminaries, we can now proceed to discuss some Korovkin-type results for L p ( X, e µ )spaces.Consider a Borel measure e µ on X and, given p ∈ [1 , + ∞ [, consider the space L p ( X, e µ )endowed with its natural norm k · k p (for more details, we refer to the Appendix (formulae (11.11)and (11.12))).Since e µ is regular, K ( X ) is dense in L p ( X, e µ ) with respect to the convergence in p -th mean (seeTheorem 11.4 of Appendix). Therefore, we get the following useful results. Corollary 8.7. Let M be a Korovkin subset of C ( X ) . Furthermore, let Y be a locally compactHausdorff space with a countable base and consider a Borel measure e µ on Y and p ∈ [1 , + ∞ [ . Let S : C ( X ) −→ L p ( Y, e µ ) be a lattice homomorphism such that K ( Y ) ⊂ S ( C ( X )) . Then S ( M ) is aKorovkin subset of L p ( Y, e µ ) .In particular, if e µ is a finite Borel measure on X , then M is also a Korovkin subset of L p ( X, e µ ) . Corollary 8.8. Let X be a compact subset of R d , d ≥ , and consider a finite Borel measure e µ on X . Then { , pr , . . . , pr d , d P i =1 pr i } is a Korovkin subset of L p ( X, e µ ) for every p ∈ [1 , + ∞ [ .If, in addition, X is contained in some sphere of R d , then { , pr , . . . , pr d } is a Korovkin subsetof L p ( X, e µ ) . Corollary 8.9. Consider < λ < λ < λ and p ∈ [1 , + ∞ [ . Then(1) { f λ , f λ , f λ } is a Korovkin subset of L p ([0 , + ∞ [) , where f λ k ( x ) := exp( − λ k x ) for every x ∈ [0 , + ∞ [ and k = 1 , , .(2) If u : R −→ ]0 , is a strictly increasing continuous function satisfying lim x → + ∞ u ( x ) = 0 and lim x →−∞ u ( x ) = 1 , then { Φ , Φ u λ , Φ u λ } is a Korovkin subset of L p ( R ) , where Φ( x ) := exp( − x ) ( x ∈ R ) . Proof. (1) Consider the lattice homomorphism S : C ([0 , → L p ([0 , + ∞ [) defined by S ( f )( x ) := exp( − λ x ) f (exp( − x )) ( f ∈ C ([0 , , x ≥ . . Altomare S maps the subset M := { , e λ − λ , e λ − λ } into { f λ , f λ , f λ } . Hence the result follows from Corollary 8.7 and Proposition 7.2, (1).(2) A similar reasoning can be used by considering now the lattice homomorphism S : C ([0 , → L p ( R ) defined by S ( f )( x ) := exp( − x ) f ( u ( x )) ( f ∈ C ([0 , , x ∈ R ) . (cid:3) We now proceed to state a result analogous to Corollary 8.7 for more general weighted functionspaces (see Section 6). Consider a continuous weight w on X and the relevant weighted space C w ( X ) defined by (6.8). Note that, if e µ is a Borel measure on X and if w − ∈ L p ( X, e µ ) for some p ∈ [1 , + ∞ [ . (8.1)then C w ( X ) ⊂ L p ( X, e µ ) and it is dense with respect to the convergence in the p -th mean. Corollary 8.10. Consider a Korovkin subset M of C w ( X ) , i.e., wM is a Korovkin subset of C ( X ) (see (6.11)), and a Borel measure e µ on X . If (8.1) holds, then M is a Korovkin subset of L p ( X, e µ ) . Proof. Consider the lattice homomorphism S : C ( X ) −→ L p ( X, e µ ) defined by S ( f ) := w − f ( f ∈ C ( X )). Then M = S ( wM ) and S ( C ( X )) = C w ( X ). Therefore, the result follows fromCorollary 8.7. (cid:3) The previous Corollary together with Corollaries 6.12 - 6.15 furnish a simple but useful methodto construct Korovkin subsets in L p ( X, e µ )-spaces.By using similar methods, we can extend some of the previous results to more general weightedfunction spaces which often occur in the applications. Some of the subsequent results are takenfrom [22, Section 2].In what follows, we shall assume that X is noncompact and we shall denote by X ∞ := X ∪ {∞} the Alexandrov one-point compactification of X (see, e.g., [30, § f ∈ C ( X ) is saidto be convergent at infinity if there exists a (unique) l ∈ R such that for any ε > K of X such that | f ( x ) − l | ≤ ε for each x ∈ X \ K . In such a case, we also writelim x →∞ f ( x ) = l . Similarly, we shall write lim x →∞ f ( x ) = ∞ to mean that for every M ≥ K of X such that f ( x ) ≥ M for every x ∈ X \ K .Given a weight w ∈ C ( X ), consider the Banach lattice C w ∗ ( X ) := { f ∈ C ( X ) | wf is convergent at infinity } (8.2)endowed with the natural (pointwise) order and the weighted norm k f k w := k wf k ∞ ( f ∈ C w ∗ ( X )) . (8.3)For every f ∈ C w ∗ ( X ), denote by T ( f ) the function on X ∞ defined by T ( f )( z ) := ( w ( z ) f ( z ) if z ∈ X, lim x →∞ w ( x ) f ( x ) if z = ∞ . (8.4) orovkin-type theorems and positive operators T ( f ) ∈ C ( X ∞ ) and k T ( f ) k ∞ = k f k ∞ . Moreover, the linear operator T : C w ∗ ( X ) → C ( X ∞ )is a lattice isomorphism whose inverse we shall denote by S : C ( X ∞ ) → C w ∗ ( X ) . Thus a subset M of C w ∗ ( X ) is a Korovkin subset of C w ∗ ( X ) if and only if T ( M ) is a Korovkin subsetof C ( X ∞ ).Consider now a Borel measure e µ on X and assume that (8.1) holds. Then C w ∗ ( X ) ⊂ L p ( X, e µ )and C w ∗ ( X ) is dense in L p ( X, e µ ) because K ( X ) ⊂ C w ∗ ( X ). Proposition 8.11. Under assumption (8.1), each Korovkin subset of C w ∗ ( X ) is a Korovkin subsetof L p ( X, e µ ) as well. Proof. The above lattice isomorphism S can be considered as a lattice homomorphism from C ( X ∞ )into L p ( X, e µ ) and its range is C w ∗ ( X ) which is dense in L p ( X, e µ ). Since H := T ( M ) is a Korovkinsubset of C ( X ∞ ), then M = S ( H ) is a Korovkin subset of L p ( X, e µ ) by Corollary 8.7. (cid:3) Below, we state some consequences of Proposition 8.11. Proposition 8.12. Let M be a subset of C w ∗ ( X ) and assume that(i) for every x , y ∈ X , x = y , there exists h ∈ L ( M ) , h ≥ , such that h ( x ) = 0 and h ( y ) > ;(ii) for every x ∈ X there exist positive functions h, k ∈ L ( M ) such that h ( x ) = 0 , lim x →∞ w ( x ) h ( x ) > , and k ( x ) > , lim x →∞ w ( x ) k ( x ) = 0 . Then M is a Korovkin subset of C w ∗ ( X ) and hence of L p ( X, e µ ) for every Borel measure e µ on X and for every p ∈ [1 , + ∞ [ satisfying (8.1). Proof. Conditions (i) and (ii) mean that T ( M ) satisfies the hypotheses of Propositions 6.4 in C ( X ∞ ). Therefore, the result follows from Propositions 6.4 and 8.11. (cid:3) Corollary 8.13. Consider a subset M of C ( X ) that separates the points of X and a strictlypositive function f ∈ C w ( X ) . Further, assume that(1) f M ∪ f M ⊂ C w ∗ ( X ) (see (6.1) and (6.2)),(2) there exists g ∈ M such that lim x →∞ w ( x ) f ( x ) g ( x ) = 0 and lim x →∞ w ( x ) f ( x ) g ( x ) > . Then { f } ∪ f M ∪ f M is a Korovkin subset of C w ∗ ( X ) and hence of L p ( X, e µ ) for every Borelmeasure e µ on X and for every p ∈ [1 , + ∞ [ satisfying (8.1). . Altomare Proof. We shall verify conditions (i) and (ii) of Proposition 8.12. Given x , y ∈ X, x = y ,there exists f ∈ M such that f ( x ) = f ( y ). Therefore, the function h := f ( f − f ( x )) ∈L ( { f } ∪ f M ∪ f M ) satisfies h ( x ) = 0 and h ( y ) > g ∈ M satisfying assumption(2), then for every x ∈ X the functions h := f ( g − g ( x )) and k := f satisfy condition (ii) ofProposition 8.12. (cid:3) Corollary 8.14. Consider f , . . . , f n ∈ C ( X ) , n ≥ , that separate the points of X , and a strictlypositive function f ∈ C w ( X ) . Further, assume that(i) f f i ∈ C w ( X ) for every i = 1 , . . . , n , as well as f n P i =1 f i ∈ C w ∗ ( X ) ;(ii) lim x →∞ w ( x ) f ( x ) n P i =1 f i ( x ) > .Then (cid:26) f , f f , . . . , f f n , f n P i =1 f i (cid:27) is a Korovkin subset C w ∗ ( X ) and hence in L p ( X, e µ ) for everyBorel measure e µ on X and for every p ∈ [1 , + ∞ [ satisfying (8.1). Proof. The proof is similar to that of Corollary 8.13, except that in this case, the function h mustbe chosen as h := f n P i =1 ( f i − f i ( x )) . (cid:3) The particular case of Corollary 8.14 where f = is worth stating separately. Corollary 8.15. Consider f , . . . , f n ∈ C ( X ) , n ≥ , that separate the points of X and w ∈ C ( X ) a weight such that(i) f i ∈ C w ( X ) for every i = 1 , . . . , n , as well as n P i =1 f i ∈ C w ∗ ( X ) ;(ii) lim x →∞ w ( x ) n P i =1 f i ( x ) > .Then (cid:26) , f , . . . , f n n P i =1 f i (cid:27) is a Korovkin subset in C w ∗ ( X ) and hence in L p ( X, e µ ) for every finiteBorel measure e µ on X and for every p ∈ [1 , + ∞ [ satisfying (8.1). A simple situation when the assumptions of Corollaries 8.14 and 8.15 are satisfied is indicatedbelow. Corollary 8.16. Consider f , . . . , f n ∈ C ( X ) , n ≥ , that separate the points of X and assumethat lim x →∞ n P i =1 f i ( x ) = + ∞ . The following statements hold:(1) If f ∈ C ( X ) is a strictly positive function, then (cid:26) f , f f , . . . , f f n , f n P i =1 f i (cid:27) is a Korovkinsubset in C w ∗ ( X ) , where w := 1 f (cid:16) n P i =1 f i (cid:17) . orovkin-type theorems and positive operators Therefore, if e µ is a Borel measure on X and ≤ p < + ∞ , and if f ∈ L p ( X, e µ ) as wellas f f i ∈ L p ( X, e µ ) for every i = 1 , . . . , n , then (cid:26) f , f f , . . . , f f n , f n P i =1 f i (cid:27) is a Korovkinsubset in L p ( X, e µ ) .(2) In particular, (cid:26) , f , . . . , f n , n P i =1 f i (cid:27) is a Korovkin subset in C w ∗ ( X ) , where w := 11 + n P i =1 f i . Therefore, if e µ is a finite Borel measure on X and ≤ p < + ∞ , and if f i ∈ L p ( X, e µ ) forevery i = 1 , . . . , n , then (cid:26) , f , . . . , f n , n P i =1 f i (cid:27) is a Korovkin subset in L p ( X, e µ ) . The following particular case of Corollary 8.16 will be used next. Corollary 8.17. Let X be an unbounded closed subset of R d , d ≥ . The following statementshold:(1) If f ∈ C ( X ) is a strictly positive function, then (cid:8) f , f pr , . . . , f pr d , f k · k (cid:9) is a Korovkinsubset in C w ∗ ( X ) , where w := 1 f (cid:16) k · k (cid:17) . Therefore, if e µ is a Borel measure on X and ≤ p < + ∞ , and if f ∈ L p ( X, e µ ) as well as k · k f ∈ L p ( X, e µ ) , then (cid:8) f , f pr , . . . , f pr d , f k · k (cid:9) is a Korovkin subset in L p ( X, e µ ) .(2) In particular, (cid:8) , pr , . . . , pr d , k · k (cid:9) is a Korovkin subset in C w ∗ ( X ) , where w := 11 + k · k . Therefore, if e µ is a finite Borel measure on X and ≤ p < + ∞ , and if k · k ∈ L p ( X, e µ ) , then (cid:8) , pr , . . . , pr d , k · k (cid:9) is a Korovkin subset in L p ( X, e µ ) . Proof. It is sufficient to apply Corollary 8.16 with f i = pr i , 1 ≤ i ≤ d . (cid:3) Below, we list some additional examples. Examples 8.18. 1) Let I be a noncompact real interval and set r := inf I ∈ R ∪ {−∞} and r := sup I ∈ R ∪ { + ∞} . Consider a strictly positive injective function f ∈ C ( I ).Assume that for some strictly positive function w ∈ C ( I ) the following properties hold:(i) lim x → r i w ( x ) f ( x ) = lim x → r i w ( x ) f ( x ) = 0 for every i = 1 , r i I ;(ii) there exists l > x → r i w ( x ) f ( x ) = l for every i = 1 , r i I . . Altomare { f , f , f } is a Korovkin subset in C w ∗ ( I ) and hence in L p ( I, e µ ) for every Borel measure e µ on I and p ∈ [1 , + ∞ [ such that 1 /w ∈ L p ( I, e µ ).The result is a direct consequence of Corollary 8.14 with n = 1 and f = f .2) Let I be an unbounded closed real interval. The following statements hold:(1) If f ∈ C ( I ) is a strictly positive function, then { f , f e , f e } is a Korovkin subset in C w ∗ ( I ), where w := 1 f (cid:16) e (cid:17) . Therefore, if e µ is a Borel measure on I and 1 ≤ p < + ∞ , and if f ∈ L p ( I, e µ ) as well as f e ∈ L p ( X, e µ ), then { f , f e , f e } is a Korovkin subset in L p ( X, e µ ).(2) In particular, { , e , e } is a Korovkin subset in C w ∗ ( I ), where w := 11 + e . Therefore, if e µ is a finite Borel measure on I and 1 ≤ p < + ∞ , and if e ∈ L p ( X, e µ ),then { , e , e } is a Korovkin subset in L p ( X, e µ ).For a rather complete list of references on Korovkin-type theorems in L p -spaces, we refer to [8,Appendix D.2.3].Next, we discuss a simple application of the above results (more precisely, of Corollary 8.17)concerning the operators G n , n ≥ 1, defined by (4.44) as G n ( f )( x ) := (cid:16) n (cid:17) d/ Z R d f ( t ) exp (cid:16) − n k t − x k (cid:17) d t ( x ∈ R d ) (8.5)for every Borel measurable function f : R d → R for which the integral (8.5) is absolutely convergent( d ≥ ϕ : R d → R be a Borel measurable strictly positive function which we assume to beintegrable with respect to the d -dimensional Lebesgue measure λ d . We denote by e µ ϕ := ϕλ d thefinite Borel measure on R d having density ϕ with respect to λ d , i.e., e µ ϕ ( B ) = Z B ϕ ( x ) d x (8.6)for every Borel subset B of R d . If p ∈ [1 , + ∞ [ and f ∈ L p ( R d , e µ ϕ ), we shall set k f k ϕ,p := (cid:16) Z R d | f ( x ) | p ϕ ( x ) d x (cid:17) /p . (8.7) Theorem 8.19. Assume that C ϕ := sup n ≥ , t ∈ R d n ϕ ( t ) (cid:16) n π (cid:17) d/ Z R d ϕ ( x ) exp (cid:16) − n k t − x k (cid:17) d x o < + ∞ . (8.8) Then for every p ∈ [1 , + ∞ [ and n ≥ and for every f ∈ L p ( R d , e µ ϕ ) , the integral (8.5) convergesabsolutely for a.e. x ∈ R d . orovkin-type theorems and positive operators Moreover G n ( f ) ∈ L p ( R d , e µ ϕ ) and k G n ( f ) k ϕ,p ≤ max { , C ϕ }k f k ϕ,p . (8.9) Finally, if pr i ∈ L p ( R d , e µ ϕ ) for every i = 1 , . . . , d , then lim n →∞ G n ( f ) = f with respect to k · k ϕ,p . (8.10) Proof. Setting, for every n ≥ K n ( x, t ) := 1 ϕ ( t ) (cid:16) n π (cid:17) d/ exp (cid:16) − n k t − x k (cid:17) , ( t, x ∈ R d ) , we get G n ( f )( x ) = Z R K n ( t, x ) f ( t ) d e µ ϕ ( t ) ( x ∈ R d ) . Moreover, if x ∈ R n is fixed, then Z R d K n ( t, x ) d e µ ϕ ( t ) = (cid:16) n π (cid:17) d/ Z R d exp (cid:16) − n k t − x k (cid:17) d t = 1and, for t ∈ R d fixed, Z R d K n ( t, x ) d e µ ϕ ( x ) ≤ C ϕ . Therefore, the first part of the statement follows from Fubini’s and Tonelli’s theorems and fromH¨older’s inequality (see also [58, Theorem 6.18]). As regards the final part, note that each pr i belongs to L p ( R d , e µ ϕ ) too because | pr i | ≤ pr i , ( i = 1 , . . . , d ).From (8.9), it follows that the sequence ( G n ) n ≥ is equibounded from L p ( R d , e µ ϕ ) into itself.Moreover, formulae (4.48)–(4.50) imply that G n ( h ) → h in L p ( R d , ˜ µ ϕ )for every h ∈ (cid:26) , pr , . . . , p d , d P i =1 pr i (cid:27) and hence the result follows from Corollary 8.17, (2). (cid:3) Remark 8.20. It is not difficult to show that condition (8.8) is satisfied, for instance, by thefunctions ϕ m ( x ) := (1 + k x k m ) − or ϕ m ( x ) = exp( − m k x k ) ( x ∈ R d ) for every m ≥ 1. For moredetails, we refer to the papers [21, Section 3], [22] and [23, Section 4] where an extension of Theorem8.19 has been established for a sequence of more general integral operators and where the operators G n , n ≥ 1, have been studied also in weighted continuous function spaces. In this section, we shall deepen the connections between Korovkin-type theorems and Stone-Weierstrass theorems, which have already been pointed out with Theorems 3.8 and 4.7.We start by giving a new proof of a generalization of Weierstrass’ theorem due to Stone ([111])by means of Theorem 6.6. We first need the following result. . Altomare Lemma 9.1. Every closed subalgebra A of C b ( X ) is a lattice subspace (i.e., | f | ∈ A for every f ∈ A ). Proof. It is well-known that t / = ∞ X n =0 (cid:18) / n (cid:19) ( t − n = lim n →∞ p n ( t )uniformly with respect to t ∈ [0 , p n ( t ) := n X k =0 (cid:18) / k (cid:19) ( t − k ( n ≥ , t ∈ [0 , . Since lim n →∞ p n (0) = 0, we also get t / = lim n →∞ q n ( t ) uniformly with respect to t ∈ [0 , , where q n := p n − p n (0), n ≥ 1. Therefore, for any f ∈ A, f = 0 , | f | = k f k (cid:18) f k f k (cid:19) / = k f k lim n →∞ q n (cid:18) f k f k (cid:19) ∈ A = A. (cid:3) Before stating the Stone approximation theorem, we recall that a subset M of C ( X ) is said to separate strongly the points of X if it separates the points of X and if for every x ∈ X thereexists f ∈ M such that f ( x ) = 0. Theorem 9.2. Let X be a locally compact Hausdorff space with a countable base and let A be aclosed subalgebra of C ( X ) that separates strongly the points of X . Then A = C ( X ) . Proof. We first prove that there exists a strictly positive function f in A . For this, note that A is separable since C ( X ) is. Denoting by { h n | n ≥ } a dense subset of A , then for every x ∈ X there exists n ≥ h n ( x ) = 0. Hence the function f := ∞ P n =1 | h n | n k h n k lies in A , because ofLemma 9.1, and is strictly positive on X .Given such a function f ∈ A , we then have that { f } ∪ f A ∪ f A ⊂ A and hence, by Theorem6.6, A is a Korovkin subspace in C ( X ).Considering f ∈ C ( X ) and ε > 0, by Theorem 8.1 there exist h , . . . , h n ∈ A , k , . . . , k n ∈ A as well as u, v ∈ C ( X ) , u ≥ , v ≥ 0, such that k u k ≤ ε, k v k ≤ ε , and, finally, (cid:13)(cid:13)(cid:13)(cid:13) inf ≤ j ≤ n k j − sup ≤ i ≤ n h i (cid:13)(cid:13)(cid:13)(cid:13) ≤ ε and sup ≤ i ≤ n h i − u ≤ f ≤ inf ≤ j ≤ n k j + v. Then f − inf ≤ j ≤ n k j ≤ v and inf ≤ j ≤ n k j − f ≤ | inf ≤ j ≤ n k j − sup ≤ i ≤ n h i | + u. Therefore, | f − inf ≤ j ≤ n k j | ≤ | sup ≤ i ≤ n h i − inf ≤ j ≤ n k j | + u + v orovkin-type theorems and positive operators k f − inf ≤ j ≤ n k j k ≤ ε which shows that f ∈ A = A (since, by Lemma 9.1, inf ≤ j ≤ n k j ∈ A ) and this completes the proof. (cid:3) As in the case of the interval [0 , 1] (see Theorem 3.8), we actually show that Theorems 6.6 and9.2 are equivalent. The next result will be very useful for our purposes. Theorem 9.3. Let X be a locally compact Hausdorff space and consider a subset M of C ( X ) suchthat the linear subspace generated by it contains a strictly positive function f ∈ C ( X ) . Given x ∈ X , denote by A ( M, x ) the subspace of all functions f ∈ C ( X ) such that µ ( f ) = f ( x ) forevery µ ∈ M + b ( X ) satisfying µ ( g ) = g ( x ) for every g ∈ { f } ∪ f M ∪ f M . Then A ( M, x ) is aclosed subalgebra of C ( X ) which contains M . Proof. A ( M, x ) is a closed subspace of C ( X ). Set M ( x ) := { x ∈ X | g ( x ) = g ( x ) for every g ∈ M } = { x ∈ X | f ( x )( g ( x ) − g ( x )) = 0 for every g ∈ M } and fix f ∈ A ( M, x ) and µ ∈ M + b ( X ) such that µ ( g ) = g ( x ) for every g ∈ { f } ∪ f M ∪ f M .In particular, µ ( f ( g − g ( x )) ) = 0 for each g ∈ M . On the other hand, if x ∈ M ( x ), then δ x ( g ) = g ( x ) for every g ∈ { f }∪ f M ∪ f M , so that f ( x ) = f ( x ) and hence f ( x ) f = f ( x ) f on M ( x ).Therefore, by Corollary 11.6 in the Appendix, we get that µ ( f ( x ) f ) = f ( x ) µ ( f ) = f ( x ) f ( x ) and hence µ ( f ) = f ( x ). Therefore, f ∈ A ( M, x ) and hence A ( M, x ) is asubalgebra of C ( X ).Finally note that, if h ∈ M, then f ( x ) h = h ( x ) f on M ( x ). Therefore, if we again fix µ ∈ M + b ( X ) such that µ ( g ) = g ( x ) for each g ∈ { f } ∪ f M ∪ f M , by applying Corollary11.6 we at once obtain that f ( x ) µ ( h ) = h ( x ) µ ( f ) = h ( x ) f ( x ), so µ ( h ) = h ( x ) and hence h ∈ A ( M, x ). (cid:3) With the help of the preceding theorem, it is easy to show this next result. Theorem 9.4. The Korovkin-type Theorem 6.6 and the Stone-Weierstrass Theorem 9.2 are equiv-alent. Proof. In light of the proof of Theorem 9.2, we have only to show that Theorem 9.2 impliesTheorem 6.6. So, consider a subset M of C ( X ) that separates the points of X and such that thelinear subspace generated by it contains a strictly positive function f ∈ C ( X ).Given x ∈ X , consider the subspace A ( M, x ) defined in the preceding theorem. By virtue ofTheorems 9.2 and 9.3, we then infer that A ( M, x ) = C ( X ). In other words, we have shown thatfor every µ ∈ M + b ( X ) and for every x ∈ X satisfying µ ( g ) = g ( x ) for every g ∈ { f }∪ f M ∪ f M ,we also have µ ( f ) = f ( x ) for every f ∈ C ( X ) and this means that { f }∪ f M ∪ f M is a Korovkinsubset in C ( X ) because of Theorem 6.1. (cid:3) For a further deepening of the relationship between Korovkin-type theorems and Stone-Weierstrasstheorems, we refer to [8, Section 4.4] and [12-13]. . Altomare 10 Korovkin-type theorems for positive projections In this last section, we shall discuss some Korovkin-type theorems for a class of positive linearprojections by showing a nontrivial application of the general theorem proved in Section 5.Consider a compact metric space X and a positive linear projection T : C ( X ) → C ( X ),i.e., T is a positive linear operator such that T ( T ( f )) = T ( f ) for every f ∈ C ( X ). We shall denoteby H T the range of T , i.e., H T := T ( C ( X )) = { h ∈ C ( X ) | T ( h ) = h } . (10.1)We shall also assume that ∈ H T (hence T ( ) = ) and that H T separates the points of X . Inthe sequel, we shall present some examples of such projections.For every x ∈ X denote by µ x ∈ M + ( X ) the Radon measure defined by µ x ( f ) := T ( f )( x ) ( f ∈ C ( X )) (10.2)and by e µ x the unique probability Borel measure on X that corresponds to µ x via the Riesz repre-sentation theorem, i.e., T ( f )( x ) = µ x ( f ) = Z X f d e µ x ( f ∈ C ( X )) . (10.3)Thus, ( e µ x ) x ∈ X is the canonical continuous selection associated with T .By the Cauchy-Schwarz inequality (2.16), for every h ∈ H T , we get | h | = | T ( h ) | ≤ p T ( ) T ( h ) = p T ( h )so that h ≤ T ( h ) . (10.4)Set Y T := { x ∈ X | T ( f )( x ) = f ( x ) for every f ∈ C ( X ) } . (10.5) Y T is closed and it is actually equal to the so-called Choquet boundary of H T and hence it is notempty (see [8, Proposition 3.3.1 and Corollary 2.6.5]). Proposition 10.1. If M is a subset of H T that separates the points of X , then Y T = { x ∈ X | T ( h )( x ) = h ( x ) for every h ∈ M } . (10.6) Moreover, if ( h n ) n ≥ is a finite or countable family of H T that separates the points of X and if theseries u := ∞ P n =1 h n is uniformly convergent on X , then u ≤ T ( u ) and Y T := { x ∈ X | T u ( x ) = u ( x ) } . (10.7) Proof. Consider x ∈ X such that T ( h )( x ) = h ( x ), i.e., µ x ( h ) = h ( x ) for each h ∈ M . Since M ⊂ H T , we also get µ x ( h ) = h ( x ) ( h ∈ M ) and { } ∪ M ∪ M is a Korovkin subset of C ( X ) byTheorem 7.1. According to Theorem 6.1, we then obtain that f ( x ) = µ x ( f ) = T ( f )( x ) for every f ∈ C ( X ). orovkin-type theorems and positive operators T ( u ) = ∞ P n =1 T ( h n ) and hence u ≤ T ( u )by (10.4). Moreover, if x ∈ X and T ( u )( x ) = u ( x ), then ∞ X n =1 (cid:16) T ( h n )( x ) − h n ( x ) (cid:17) = 0 . Hence, on account of (10.4), T ( h n )( x ) = h n ( x ) for every n ≥ x ∈ Y T by (10.6). (cid:3) Remark 10.2. Note that there always exists a countable family ( h n ) n ≥ of H T that separates thepoints of X and such that the series ∞ P n =1 h n is uniformly convergent on X . Actually, since C ( X ) isseparable, H T is separable as well so that, considering a countable dense family ( ϕ n ) n ≥ of H T , itis enough to put h n := ϕ n n (1+ k ϕ n k ) ( n ≥ . Theorem 10.3. If M is a subset of H T that separates the points of X , then H T ∪ M is a Korovkin subset for T. Moreover, if u ∈ C ( X ) satisfies u ≤ T ( u ) and if (10.7) holds, then H T ∪ { u } is a Korovkin subset for T. In particular, the above statement holds for u = ∞ P n =1 h n , where ( h n ) n ≥ is an arbitrary sequencein H T that separates the points of X and such that the series ∞ P n =1 h n is uniformly convergent on X . Proof. In order to apply Theorem 5.5, consider µ ∈ M + ( X ) and x ∈ X such that µ ( h ) = T ( h )( x ) = ( µ x ( h )) for every h ∈ H T ∪ M . Accordingly, if h ∈ M , µ ( T ( h ) − h ) = T ( h )( x ) − µ x ( h ) = 0and T ( h ) − h ≥ 0. Therefore, by using (10.6) and by applying Theorem 11.5 and Corollary 11.6of the Appendix, we see that, for every f ∈ C ( X ), µ ( T ( f )) = µ ( f ) because T f = f on Y T . Hence µ ( f ) = µ ( T ( f )) = µ x ( T f ) = T f ( x ) and this finishes the proof.A similar reasoning can be used to show the second part of the statement by using (10.7) insteadof (10.6). (cid:3) Theorem 10.3 is due to Altomare ([3], [5]; see also [8, Section 3.3]). For an extension to so calledadapted spaces, we refer to [9]. Below, we show some examples. Example 10.4. Consider the d -dimensional simplex K d of R d , d ≥ 1, defined by (4.38) and thepositive projection T d : C ( K p ) → C ( K p ) defined by T d ( f )( x ) := (cid:16) − d X i =1 x i (cid:17) f (0) + d X i =1 x i f ( a i ) (10.8)( f ∈ C ( K d ), x = ( x i ) ≤ i ≤ d ∈ K d ), where a i := ( δ ij ) ≤ j ≤ d for every i = 1 , . . . , d . . Altomare H T d is the subspace A ( K d ) (see Section 7, formula (7.1)) which in turn is generatedby M := { , pr , . . . , pr d } , and hence, by Theorem 10.3, ( , pr , . . . , pr d , d X i =1 pr i ) is a Korovkin subset for T d . (10.9)More generally, if K is an infinite-dimensional Bauer simplex (see, e.g., [8, Section 1.5, p. 59]),then there exists a unique linear positive projection T on C ( K ) whose range is A ( K ). In this case,for every strictly convex u ∈ C ( K ), we get that A ( K ) ∪ { u } is a Korovkin subset for T (see [8,Corollary 3.3.4]). Example 10.5. Consider the hypercube Q d := [0 , d of R d , d ≥ 1, and the positive projection S d : C ( Q d ) → C ( Q d ) defined by S d ( f )( x ) := X h ,...,h d =0 x h (1 − x ) − h · · · x h d d (1 − x d ) − h d f ( b h ,...,h d ) (10.10)where b h ,...,h d := ( δ h i ) ≤ i ≤ d ( h , . . . , h d ∈ { , } ) . In this case, H S d is the subspace of C ( Q d ) generated by { } ∪ { Q i ∈ J pr i | J ⊂ { , . . . , d }} . Therefore, by Theorem 10.3, we obtain that { } ∪ (Y i ∈ J pr i | J ⊂ { , . . . , d } ) ∪ ( d X i =1 pr i ) is a Korovkin subset for S d . (10.11)For an extension of the above result, see [8, Corollary 3.3.9]. Example 10.6. Consider a bounded open subset Ω of R d , d ≥ 2, which we assume to be regularin the sense of potential theory (see, e.g., [68, Section 8.3] or [8, pp. 125-128]) (for instance, eachconvex open subset of R d is regular). Denote by H (Ω) the subspace of all u ∈ C (Ω) that areharmonic on Ω.By the regularity of Ω, it follows that for every f ∈ C (Ω) there exists a unique u f ∈ H (Ω) suchthat u f | ∂ Ω = f | ∂ Ω , i.e., u f is the unique solution of the Dirichlet problem △ u := d P i =1 ∂ u∂x i = 0 on Ω ,u | ∂ Ω = f | ∂ Ω ( u ∈ C (Ω) ∩ C (Ω)) . (10.12)Then the Poisson operator T : C (Ω) −→ C (Ω) defined by T ( f ) := u f ( f ∈ C (Ω)) (10.13)is a positive projection whose range is H (Ω).Therefore, from Theorem 10.3, we get that H (Ω) ∪ ( d X i =1 pr i ) is a Korovkin subset for T. (10.14) orovkin-type theorems and positive operators K of a locally convex space. For every f ∈ C ( K ), z ∈ K and α ∈ [0 , f z,α ∈ C ( K )the function defined by f z,α ( x ) := f ( αx + (1 − α ) z ) ( x ∈ K ) . (10.15)Consider a positive linear projection T : C ( K ) −→ C ( K ), T different from the identity operator,and assume that A ( K ) ⊂ H T := T ( C ( K )) , (10.16)i.e., T ( u ) = u for every u ∈ A ( K ) , (10.17)and h z,α ∈ H T for every h ∈ H T , z ∈ K, ≤ α ≤ . (10.18)Here, once again, we use the symbol A ( K ) to designate the subspace of all continuous affinefunctions on K (see (7.1)). Moreover, note that in all the examples 10.4-10.6, assumptions (10.16)–(10.18) are satisfied.Moreover, from Theorem 10.3, it also follows that H T ∪ A ( K ) is a Korovkin subset for T. (10.19)Consider the sequence of Bernstein-Schnabl operators ( B n ) n ≥ associated with T according to (7.6)and (7.7), i.e., associated with the continuous selection ( e µ x ) x ∈ X defined by (10.3).Note that, if T is the projection T p defined by (10.8) (resp., the projection S p defined by (10.10)),then the corresponding Bernstein-Schnabl operators are the Bernstein operators (4.39) (resp., theBernstein operators (4.41)). By Theorem 7.5, we already know thatlim n →∞ B n ( f ) = f uniformly on K ( f ∈ C ( K )) . (10.20)In this particular case, we can also study the iterates of the operators B n .For every n, m ≥ 1, we set B mn := ( B n if m = 1 ,B n ◦ B m − n if m ≥ . (10.21)From (10.18), it follows that, if h ∈ H T , then B n ( h ) = h and hence B mn ( h ) = h = T ( h ) . (10.22)On the other hand, if u ∈ A ( K ), then by applying an induction argument it is not difficult toobtain from (7.13) that B n ( u ) = (cid:16) − (cid:16) n − n (cid:17) m (cid:17) T ( u ) + (cid:16) n − n (cid:17) m u . (10.23)Now, we can easily prove the next result. . Altomare Theorem 10.7. Let f ∈ C ( K ) . Then(1) lim n →∞ B mn ( f ) = f uniformly on K for every m ≥ .(2) lim m →∞ B mn ( f ) = T ( f ) uniformly on K for every n ≥ .(3) If ( k ( n )) n ≥ is a sequence of positive integers, then lim n →∞ B k ( n ) n ( f ) = ( f uniformly on K if k ( n ) n → ,T ( f ) uniformly on K if k ( n ) n → + ∞ . Proof. It is sufficient to apply Corollary 7.3 and statement (10.19) taking (10.22) and (10.23) intoaccount as well as the elementary formula (cid:16) n − n (cid:17) m = exp (cid:16) m log (cid:16) − n (cid:17)(cid:17) ( n, m ≥ . (cid:3) It is worth remarking that under some additional assumptions on T , the sequence ( B k ( n ) n ( f )) n ≥ ( f ∈ C ( K )) converges uniformly also when k ( n ) n −→ t ∈ ]0 , + ∞ [. More precisely, for every t ≥ T ( t ) : C ( K ) −→ C ( K ) such that for every sequence ( k ( n )) n ≥ of positive integers satisfying k ( n ) n −→ t , and for every f ∈ C ( K ), T ( t ) f = lim n →∞ B k ( n ) n ( f ) . (10.24)Moreover, the family ( T ( t )) t ≥ is a strongly continuous semigroup of operators whose generator( A, D ( A )) is the closure of the operator ( Z, D ( Z )) where D ( Z ) := n u ∈ C ( K ) | lim n →∞ n ( B n ( u ) − u ) exists in C ( K ) o (10.25)and, for every u ∈ D ( Z ) ⊂ D ( A ), A ( u ) = Z ( u ) = lim n →∞ n ( B n ( u ) − u ) . (10.26)If K is a subset of R d , d ≥ 1, with nonempty interior and if T maps the subspace of all polynomialsof degree m into itself for every m ≥ 1, then C ( K ) ⊂ D ( Z ) ⊂ D ( A ) and, for every u ∈ C ( K ), Au ( x ) = Zu ( x ) = 12 d X i,j =1 α ij ( x ) ∂ u ( x ) ∂x i ∂x i (10.27)( x = ( x i ) ≤ i ≤ d ), where for every i, j = 1 , . . . , d , α ij ( x ) := T ( pr i pr j )( x ) − x i x j . (10.28)The differential operator (10.27) is an elliptic second order differential operator which degenerateson the subset Y T defined by (10.5) (which contains all the extreme points of K ). orovkin-type theorems and positive operators u ∈ D ( A ), the (abstract) Cauchy problem ∂u∂t ( x, t ) = A ( u ( · , t ))( x ) x ∈ K, t ≥ ,u ( x, 0) = u ( x ) x ∈ K,u ( · , t ) ∈ D ( A ) t ≥ , (10.29)has a unique solution u : K × [0 , + ∞ [ → R given by u ( x, t ) = T ( t )( u )( x ) = lim n →∞ B k ( n ) n ( u )( x ) ( x ∈ K, t ≥ 0) (10.30)and the limit is uniform with respect to x ∈ K , where k ( n ) n → t .These results were discovered by Altomare ([5], see also [8, Sections 6.2 and 6.3]) and openedthe door to a series of researches whose main aims are, first, the approximation of the solutions ofinitial boundary differential problem associated with degenerate evolution equations (like (10.29))by means of iterates of positive linear operators (like (10.30)), and, second, both a numerical anda qualitative analysis of the solutions by means of formula (10.30).These researches are documented in several papers. Here, we content ourselves to cite, otherthan Chapter 6 of [8], also the papers [6], [17] and [18] and the references therein. 11 Appendix: A short review of locally compact spaces and of some continuousfunction spaces on them For the convenience of the reader, in this Appendix, we collect some basic definitions and resultsconcerning locally compact Hausdorff spaces, some continuous function spaces and Radon measureson them. For more details, we refer the reader to Chapter IV of [30] or to Chapter 3 of [54].We start by recalling that a topological space X is said to be compact if every open cover of X has a finite subcover. A subset of a topological space is said to be compact if it is compact in therelative topology. A topological space is said to be locally compact if each of its points possessesa compact neighborhood.Actually, if X is locally compact and Hausdorff (i.e., for every pair of distinct points x , x ∈ X there exist neighborhoods U and U of x and x , respectively, such that U ∩ U = ∅ ), then eachpoint of X has a fundamental system of compact neighborhoods.Every compact space is locally compact. The spaces R d , d ≥ 1, are fundamental examplesof (noncompact) locally compact spaces. Furthermore, if X is locally compact, then every opensubset of X and every closed subset of X , endowed with the relative topology, is locally compact.More generally, a subset of a locally compact Hausdorff space, endowed with the relative topology,is locally compact if and only if it is the intersection of an open subset of X with a closed subsetof X (see [54, Corollary 3.3.10]). Therefore, every real interval is locally compact.A topological space X is said to be metrizable if its topology is induced by a metric on X .In this case, we say that X is complete if such a metric is complete. Note that every compactmetrizable space is complete and separable , i.e., it contains a dense countable subset.A special role in the measure theory on topological spaces (and in the Korovkin-type approxi-mation theory) is played by locally compact Hausdorff spaces with a countable base (or basis),i.e., with a countable family of open subsets such that every open subset is the union of somesubfamily of it. Such spaces are metrizable, complete and separable. Actually, a metrizable space . Altomare R d , d ≥ 1, and each open or closedsubset of them are locally compact Hausdorff spaces with a countable base.From now on, X will stand for a fixed locally compact Hausdorff space. We denote by K ( X )the linear space of all real-valued continuous functions f : X −→ R whose (closed) supportsupp( f ) := { x ∈ X | f ( x ) = 0 } is compact. K ( X ) is a lattice subspace of C b ( X ) and it coincides with C ( X ) if X is compact.The next result shows that there are sufficiently many functions in K ( X ). (For a proof, see [30,Corollary 27.3].) Theorem 11.1. ( Urysohn’s lemma ) For every compact subset K of X and for every open subset U containing K , there exists ϕ ∈ K ( X ) such that ≤ ϕ ≤ , ϕ = 1 on K and supp( ϕ ) ⊂ U (andhence ϕ = 0 on X \ U ). Another fundamental function space is the space C ( X ) which is defined as the closure of K ( X )in C b ( X ) with respect to the sup norm, in symbols C ( X ) := K ( X ) . Thus, C ( X ) is a closed linear subspace of C b ( X ) and hence, endowed with the sup-norm, is aBanach space.By means of Urysohn’s lemma, it is not difficult to prove the following characterization offunctions lying in C ( X ). Theorem 11.2. Assume that X is not compact. For a function f ∈ C ( X ) , the following statementsare equivalent:(i) f ∈ C ( X ) ;(ii) { x ∈ X | | f ( x ) | ≥ ε } is compact for every ε > ;(iii) for every ε > there exists a compact subset K of X such that | f ( x ) | ≤ ε for every x ∈ X \ K . Because of the preceding theorem, the functions lying in C ( X ) are said to vanish at infinity. If X is compact, then C ( X ) = C ( X ). Moreover, C ( X ) is a lattice subspace of C b ( X ) and, endowedwith the sup-norm, is separable provided X has a countable base.Another characterization of functions in C ( X ) involves sequences of points of X that convergeto the point at infinity of X . More precisely, assuming that X is noncompact, a sequence ( x n ) n ≥ in X is said to converge to the point at infinity of X if for every compact subset K of X thereexists ν ∈ N such that x n ∈ X \ K for every n ≥ ν . For any such sequence and for every f ∈ C ( X ),we then have lim n →∞ f ( x n ) = 0 . (11.1)Conversely, a function f ∈ C ( X ) satisfying (11.1) for every sequence ( x n ) n ≥ converging to thepoint at infinity of X necessarily lies in C ( X ) provided that X is countable at infinity, i.e., it is theunion of a sequence of compact subsets of X . Note also that X is countable at infinity if and onlyif there exists f ∈ C ( X ) such that f ( x ) > x ∈ X . Moreover, if X has a countablebasis, then X is countable at infinity. orovkin-type theorems and positive operators Radon measures . Actually, we shall only need to handle positive bounded Radon measures which are, by definition, positive linear functionals on C ( X ). The set of all of them will be denotedby M + b ( X ).Every µ ∈ M + b ( X ), that is, every positive linear functional µ : C ( X ) −→ R , is continuous(with respect to the sup-norm), and its norm k µ k := sup {| µ ( f ) | | f ∈ C ( X ) , | f | ≤ } (11.2)is also called the total mass of µ .A simple example of bounded positive Radon measure is furnished by the Dirac measure ata point a ∈ X , which is defined by δ a ( f ) := f ( a ) ( f ∈ C ( X )) . (11.3)A positive linear combination of Dirac measures is called a (positive) discrete measure .In other words, a Radon measure µ ∈ M + b ( X ) is discrete if there exist finitely many points a , . . . , a n ∈ X, n ≥ 1, and finitely many positive real numbers λ , . . . , λ n such that µ = n X i =1 λ i δ a i , (11.4)i.e., µ ( f ) = n X i =1 λ i f ( a i ) for every f ∈ C ( X ) . (11.5)In this case, k µ k = n P i =1 λ i and µ is also said to be supported on { a , . . . , a n } .There is a strong relationship between positive bounded Radon measures and (positive) finiteBorel measures on X . In order to briefly describe it, we recall that the Borel σ -algebra in X is,by definition, the σ -algebra generated by the system of all open subsets of X . It will be denotedby B ( X ) and its elements are called Borel subsets of X . Open subsets, closed subsets and compactsubsets of X are Borel subsets.A Borel measure e µ on X is, by definition, a measure e µ : B ( X ) → [0 , + ∞ ] such that e µ ( K ) < + ∞ for every compact subset K of X. (11.6)Every finite measure e µ on B ( X ), i.e., e µ ( X ) < + ∞ , is a Borel measure.A measure e µ : B ( X ) −→ [0 , + ∞ ] is said to be inner regular if e µ ( B ) = sup { e µ ( K ) | K ⊂ B, K compact } for every B ∈ B ( X ) (11.7)and outer regular if e µ ( B ) = inf { e µ ( U ) | B ⊂ U, U open } for every B ∈ B ( X ) . (11.8)A measure e µ is said to be regular if it is both inner regular and outer regular. The Lebesgue-Borelmeasure on R d , d ≥ 1, is regular. Actually, if X has a countable base, then every Borel measureson X is regular ([30, Theorem 29.12]). . Altomare e µ is a finite measure on B ( X ), then every f ∈ C b ( X ) is e µ -integrable. Therefore, we canconsider the positive bounded Radon measure I e µ on X defined by I e µ ( f ) := Z X f d e µ ( f ∈ C ( X )) . (11.9)Then k I e µ k ≤ e µ ( X ), and, if e µ is inner regular, k I e µ k = e µ ( X ).As a matter of fact, formula (11.9) describes all the positive bounded Radon measures on X asthe following fundamental result shows (see [30, Section 29]). Theorem 11.3. ( Riesz representation theorem ) If µ ∈ M + b ( X ) , then there exists a uniquefinite and regular Borel measure e µ on X such that µ ( f ) = Z X f d e µ for every f ∈ C ( X ) . Moreover, k µ k = e µ ( X ) . Another noteworthy and useful result concerning regular Borel measures is shown below. Con-sider a measure e µ on B ( X ) and p ∈ [1 , + ∞ [. As usual, we shall denote by L p ( X, e µ ) the linearsubspace of all B ( X )-measurable functions f : X −→ R such that | f | p is e µ -integrable.If f ∈ L p ( X, e µ ), it is customary to set N p ( f ) := (cid:18)Z X | f | p d e µ (cid:19) /p . (11.10)The functional N p : L p ( X, e µ ) → R is a seminorm and the convergence with respect to it is theusual convergence in p th -mean . Setting N := { f ∈ L p ( X, e µ ) | N p ( f ) = 0 } = { f ∈ F ( X ) | f is B ( X ) measurable and f = 0 e µ a.e. } , the quotient linear space L p ( X, e µ ) := L p ( X, e µ ) / N (11.11)endowed with the norm k e f k p := N p ( f ) ( e f ∈ L p ( X, e µ )) , (11.12)is a Banach space. (Here, e f := { g ∈ L p ( X, e µ ) | f = g e µ a.e. } . ) Note that, if e µ is a Borel measure,then K ( X ) ⊂ L p ( X, e µ ) for every p ∈ [1 , + ∞ [.If e µ is also regular, we can say much more (for a proof of the next result, see [30, Theorem29.14]). Theorem 11.4. If e µ is a regular Borel measure on X , then, for every p ∈ [1 , + ∞ [ , the space K ( X ) is dense in L p ( X, e µ ) with the respect to convergence in the p th -mean (and hence in L p ( X, e µ ) withrespect to k · k p ). Next, we discuss a characterization of discrete Radon measures. Usually this characterizationis proved by using the notion of support of Radon measures (see, e.g., [36, Chapter III, Section 2]and [39, Vol. I, Section 11]). Below, we present a simple and direct proof.We start with the following result which is important in its own right. orovkin-type theorems and positive operators Theorem 11.5. Let µ ∈ M + b ( X ) and consider a closed subset Y of X such that µ ( ϕ ) = 0 for every ϕ ∈ K ( X ) , supp( ϕ ) ⊂ X \ Y. (11.13) Then µ ( f ) = µ ( g ) for every f, g ∈ C ( X ) such that f = g on Y . Proof. It suffices to show that, if f ∈ C ( X ) and f = 0 on Y , then µ ( f ) = 0. Consider such afunction f ∈ C ( X ) and, given ε > 0, set U := { x ∈ X | | f ( x ) | < ε } . Hence, by Theorem 11.2, X \ U is compact and X \ U ⊂ X \ Y .By Urysohn’s lemma (Theorem 11.1), there exists ϕ ∈ K ( X ), 0 ≤ ϕ ≤ 1, such that ϕ = 1 on X \ U and supp( ϕ ) ⊂ X \ Y . In particular, supp( f ϕ ) ⊂ supp( ϕ ) ⊂ X \ Y and hence µ ( f ϕ ) = 0.Therefore, | µ ( f ) | = | µ ( f ) − µ ( f ϕ ) | ≤ k µ kk f (1 − ϕ ) k ≤ k µ k ε, because k f (1 − ϕ ) k ≤ ε as we now confirm. For every x ∈ X , we have, indeed, | f ( x )(1 − ϕ ( x )) | = 0if x U and, if x ∈ U, | f ( x )(1 − ϕ ( x )) | ≤ | f ( x ) | ≤ ε . Since ε > µ ( f ) = 0. (cid:3) We point out that there always exists a closed subset Y of X satisfying (11.13). The smallest ofthem is called the support of the measure µ (see the references before Theorem 11.5). An importantexample of a subset Y satisfying (11.13) is given below. Corollary 11.6. Let µ ∈ M + b ( X ) and consider an arbitrary family ( f i ) i ∈ I of positive functions in C ( X ) such that µ ( f i ) = 0 for every i ∈ I . Then the subset Y := { x ∈ X | f i ( x ) = 0 for every i ∈ I } satisfies (11.13).Therefore, if f, g ∈ C ( X ) and if f ( x ) = g ( x ) for every x ∈ Y, then µ ( f ) = µ ( g ) . Proof. Consider ϕ ∈ K ( X ) such thatsupp( ϕ ) ⊂ X \ Y = { x ∈ X | there exists i ∈ I with f i ( x ) > } . By using a compactness argument, we then find a finite subset J of I such thatsupp( ϕ ) ⊂ [ i ∈ J { x ∈ X | f i ( x ) > } . If we set α := min (cid:26) P i ∈ J f i ( x ) | x ∈ supp( ϕ ) (cid:27) > 0, we then obtain | ϕ | ≤ k ϕ k α X i ∈ J f i and hence µ ( f ) = 0. (cid:3) By means of the previous result, it is easy to reach the announced characterization of discreteRadon measures. . Altomare Theorem 11.7. Given µ ∈ M + b ( X ) and different points a , . . . , a n ∈ X, n ≥ , the followingstatements are equivalent:(i) There exist λ , . . . , λ n ∈ [0 , + ∞ [ such that µ = n P i =1 λ i δ a i (see (11.5));(ii) if ϕ ∈ K ( X ) and supp( ϕ ) ∩ { a , . . . , a n } = ∅ , then µ ( ϕ ) = 0 ;(iii) for every x ∈ X \{ a , . . . , a n } there exists f ∈ C ( X ) , f ≥ , such that f ( x ) > , f ( a i ) = 0 for each i = 1 , . . . , n , and µ ( f ) = 0 . Proof. (i) ⇒ (ii). It is obvious.(ii) ⇒ (iii). If x ∈ X \{ a , . . . , a n } , by Urysohn’s lemma, we can choose ϕ ∈ K ( X ), 0 ≤ ϕ ≤ ϕ ( x ) = 1 and supp( ϕ ) ⊂ X \{ a , . . . , a n } so that µ ( f ) = 0.(iii) ⇒ (ii). Consider ϕ ∈ K ( X ) such that supp( ϕ ) ∩ { a , . . . , a n } = ∅ . By hypothesis, for every x ∈ supp( ϕ ), there exists f x ∈ C ( X ), f x ≥ 0, such that f x ( x ) > f x ( a i ) = 0 for every i = 1 , . . . , n ,and µ ( f x ) = 0.Since supp( ϕ ) is compact, there exist x , . . . , x p ∈ supp( ϕ ) such thatsupp( ϕ ) ⊂ p [ k =1 { x ∈ X | f x k ( x ) > } . Therefore, the function f := p P k =1 f x k ∈ C ( X ) is positive, it does not vanish at any point of supp( ϕ ),and µ ( f ) = 0.If we set m := min { f ( x ) | x ∈ supp( ϕ ) } > 0, it is immediate to verify that m | ϕ | ≤ k ϕ k f andhence µ ( ϕ ) = 0.(ii) ⇒ (i). For every j = 1 , . . . , n , consider ϕ j ∈ K ( X ) such that 0 ≤ ϕ j ≤ ϕ j ( a j ) = 1 and ϕ j ( a i ) = 0 for each i = 1 , . . . , n , i = j .If f ∈ C ( X ), then f = n X i =1 f ( a i ) ϕ i on { a , . . . , a n } . On the other hand, the subset { a , . . . , a n } satisfies (11.13) and hence, by Theorem 11.5, µ ( f ) = n X i =1 f ( a i ) µ ( ϕ i ) = n X i =1 λ i f ( a i )where λ i := µ ( ϕ i ) ≥ i = 1 , . . . , n ) and this completes the proof. (cid:3) We end the Appendix by discussing some aspects of vague convergence for Radon measures.For more details, we refer, e.g., to [30, § 30] or to [39, Vol. I, § µ n ) n ≥ in M + b ( X ) is said to converge vaguely to µ ∈ M + b ( X ) iflim n →∞ µ n ( f ) = µ ( f ) for every f ∈ C ( X ) . (11.14)Thus, (11.14) simply means that µ n → µ weakly in the dual space of C ( X ).If, in addition, X has a countable base, then C ( X ) is separable and hence, by Banach’s theorem,the unit ball of the dual of C ( X ) is weakly sequentially compact. Therefore Theorem 11.8. 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