Quantitative estimates in approximation by Bernstein-Durrmeyer-Choquet operators with respect to monotone and submodular set functions
aa r X i v : . [ m a t h . C A ] N ov Quantitative estimates in approximation byBernstein-Durrmeyer-Choquet operators withrespect to monotone and submodular setfunctions
Sorin G. GalDepartment of Mathematics and Computer Science,University of Oradea,Universitatii Street No.1, 410087, Oradea, RomaniaE-mail: [email protected]
Abstract
For the qualitative results of pointwise and uniform approximationobtained in [10], we present general quantitative estimates in terms of themodulus of continuity and in terms of a K -functional, for the generalizedmultivariate Bernstein-Durrmeyer operator M n, Γ n,x , written in terms ofthe Choquet integral with respect to a family of monotone and submodularset functions, Γ n,x , on the standard d -dimensional simplex. When Γ n,x reduces to two elements, one a Choquet submodular set function and theother one a Borel measure, for suitable modified Bernstein-Durrmeyeroperators, univariate L p -approximations, p ≥
1, with estimates in termsof a K -functional are proved. In the particular cases when d = 1 andthe Choquet integral is taken with respect to some concrete possibilitymeasures, the pointwise estimate in terms of the modulus of continuityis detailed. Some simple concrete examples of operators improving theclassical error estimates are presented. Potential applications to practicalmethods dealing with data, like learning theory and regression models,also are mentioned. AMS 2000 Mathematics Subject Classification : 41A36, 41A25, 28A12,28A25.
Keywords and phrases : Bernstein-Durrmeyer-Choquet operator, pos-sibility measure, monotone and submodular set function, Choquet integral,quantitative estimates, pointwise approximation, uniform approximation, L p -approximation, modulus of continuity, K -functional.1 Introduction
The approximation properties of the multivariate Bernstein-Durrmeyer linearoperator defined with respect to a nonnegative, bounded Borel measure µ : B S d → R + , by M n,µ ( f )( x )= X | α | = n R S d f ( t ) B α ( t ) dµ ( t ) R S d B α ( t ) dµ ( t ) · B α ( x ) := X | α | = n c ( α, µ ) · B α ( x ) , x ∈ S d , n ∈ N , (1)where B S d denotes the sigma algebra of all Borel measurable subsets in thepower set P ( S d ) and f is supposed to be µ -integrable on the standard simplex S d = { ( x , ..., x d ); 0 ≤ x , ..., x d ≤ , ≤ x + ... + x d ≤ } , were studied in, e.g., the recent papers [4], [1], [2], [3] and [13].Note that in (1), it is used the notation B α ( x ) = n ! α ! · α ! · ... · α d ! (1 − x − x − ... − x d ) α · x α · ... · x α d d =: n ! α ! · α ! · ... · α d ! · P α ( x ) , where α = ( α , α , ..., α d ), α j ∈ N S { } , j = 0 , ..., d , | α | = α + α + ... + α d = n .In the very recent paper [10], we have proved that the approximation resultsin the above mentioned papers remain valid for the more general case when µ is a monotone, normalized and submodular set function on S d and the integralsused in (1) are the nonlinear Choquet integrals with respect to µ .The main goal of this paper is to obtain quantitative estimates in terms ofthe modulus of continuity and in terms of some K -functionals, for the point-wise and uniform approximation obtained in [10] and for the univariate L p -approximation, p ≥
1, in the case of more general multivariate Bernstein-Durrmeyer polynomial operators defined by M n, Γ n,x ( f )( x ) = X | α | = n c ( α, µ n,α,x ) · B α ( x ) , x ∈ S d , n ∈ N , (2)where c ( α, µ n,α,x ) = ( C ) R S d f ( t ) B α ( t ) dµ n,α,x ( t )( C ) R S d B α ( t ) dµ n,α,x ( t ) = ( C ) R S d f ( t ) P α ( t ) dµ n,α,x ( t )( C ) R S d P α ( t ) dµ n,α,x ( t )and for every n ∈ N and x ∈ S d , Γ n,x = ( µ n,α,x ) | α | = n is a family of bounded,monotone, submodular and strictly positive set functions on B S d .Note that if Γ n,x reduces to one element (i.e. µ n,α,x = µ for all n , x and α ),then the operator given by (2) reduces to the operator considered in [10].The plan of the paper is as follows. Section 2 contains some preliminarieson possibility theory and on Choquet integral. In Section 3, general quan-titative estimates in terms of the modulus of continuity and in terms of a2 -functional for the pointwise and uniform approximation by the operators M n, Γ n,x ( f )( x ) defined by (2) are obtained. Also, when Γ n,x reduces to twoelements, one a Choquet submodular set function and the other one a Borelmeasure, for suitable modified Bernstein-Durrmeyer-Choquet operators, uni-variate L p -approximations, p ≥
1, with quantitative estimates in terms of a K -functional are presented. Finally, in Section 4, in the particular case when d = 1 and the Choquet integrals are taken with respect to some concrete possi-bility measures, the pointwise estimate in terms of the modulus of continuity isdetailed. Also, some concrete example of operators improving the classical errorestimates are presented and potential applications to practical methods dealingwith data are mentioned. Firstly, we present a few known concepts in possibility theory useful for the nextconsiderations. For details, see, e.g., [8].
Definition 2.1.
For the non-empty set Ω, denote by P (Ω) the family of allsubsets of Ω.(i) A function λ : Ω → [0 ,
1] with the property sup { λ ( s ); s ∈ Ω } = 1, iscalled possibility distribution on Ω.(ii) A possibility measure is a set function P : P (Ω) → [0 , P ( ∅ ) = 0, P (Ω) = 1 and P ( S i ∈ I A i ) = sup { P ( A i ); i ∈ I } for all A i ⊂ Ω,and any I , an at most countable family of indices. Note that if A, B ⊂ Ω, A ⊂ B , then the last property easily implies that P ( A ) ≤ P ( B ) and that P ( A S B ) ≤ P ( A ) + P ( B ).Any possibility distribution λ on Ω, induces the possibility measure P λ : P (Ω) → [0 , P λ ( A ) = sup { λ ( s ); s ∈ A } , for all A ⊂ Ω(see, e.g., [8], Chapter 1).Some known concepts and results concerning the Choquet integral can besummarized by the following.
Definition 2.2.
Suppose Ω = ∅ and let C be a σ -algebra of subsets in Ω.(i) (see, e.g., [17], p. 63) The set function µ : C → [0 , + ∞ ] is called amonotone set function (or capacity) if µ ( ∅ ) = 0 and µ ( A ) ≤ µ ( B ) for all A, B ∈C , with A ⊂ B . Also, µ is called submodular if µ ( A [ B ) + µ ( A \ B ) ≤ µ ( A ) + µ ( B ) , for all A, B ∈ C .µ is called bounded if µ (Ω) < + ∞ and normalized if µ (Ω) = 1.(ii) (see, e.g., [17], p. 233, or [6]) If µ is a monotone set function on C andif f : Ω → R is C -measurable (that is, for any Borel subset B ⊂ R it follows f − ( B ) ∈ C ), then for any A ∈ C , the concept of Choquet integral is defined by( C ) Z A f dµ = Z + ∞ µ (cid:16) F β ( f ) \ A (cid:17) dβ + Z −∞ h µ (cid:16) F β ( f ) \ A (cid:17) − µ ( A ) i dβ, where we used the notation F β ( f ) = { ω ∈ Ω; f ( ω ) ≥ β } . Notice that if f ≥ A , then in the above formula we get R −∞ = 0.3he function f will be called Choquet integrable on A if ( C ) R A f dµ ∈ R .In what follows, we list some known properties of the Choquet integral. Remark 2.3. If µ : C → [0 , + ∞ ] is a monotone set function, then thefollowing properties hold :(i) For all a ≥ C ) R A af dµ = a · ( C ) R A f dµ (if f ≥ f is of arbitrary sign, then see, e.g.,[7], p. 64, Proposition 5.1, (ii)).(ii) For all c ∈ R and f of arbitrary sign, we have (see, e.g., [17], pp. 232-233,or [7], p. 65) ( C ) R A ( f + c ) dµ = ( C ) R A f dµ + c · µ ( A ).If µ is submodular too, then for all f, g of arbitrary sign and lower bounded,we have (see, e.g., [7], p. 75, Theorem 6.3)( C ) Z A ( f + g ) dµ ≤ ( C ) Z A f dµ + ( C ) Z A gdµ. (iii) If f ≤ g on A then ( C ) R A f dµ ≤ ( C ) R A gdµ (see, e.g., [17], p. 228,Theorem 11.2, (3) if f, g ≥ f, g are of arbitrary sign).(iv) Let f ≥
0. If A ⊂ B then ( C ) R A f dµ ≤ ( C ) R B f dµ. In addition, if µ isfinitely subadditive, then( C ) Z A S B f dµ ≤ ( C ) Z A f dµ + ( C ) Z B f dµ. (v) It is immediate that ( C ) R A · dµ ( t ) = µ ( A ).(vi) The formula µ ( A ) = γ ( M ( A )), where γ : [0 , → [0 ,
1] is an increasingand concave function, with γ (0) = 0, γ (1) = 1 and M is a probability measure(or only finitely additive) on a σ -algebra on Ω (that is, M ( ∅ ) = 0, M (Ω) = 1 and M is countably additive), gives simple examples of normalized, monotone andsubmodular set functions (see, e.g., [7], pp. 16-17, Example 2.1). For example,we can take γ ( t ) = √ t .If the above γ function is increasing, concave and satisfies only γ (0) = 0,then for any bounded Borel measure m , µ ( A ) = γ ( m ( A )) gives a simple exampleof bounded, monotone and submodular set function.Note that any possibility measure µ is normalized, monotone and submodu-lar. Indeed, the axiom µ ( A S B ) = max { µ ( A ) , µ ( B ) } implies the monotonicity,while the property µ ( A T B ) ≤ min { µ ( A ) , µ ( B ) } implies the submodularity.(vii) If µ is a countably additive bounded measure, then the Choquet integral( C ) R A f dµ reduces to the usual Lebesgue type integral (see, e.g., [7], p. 62, or[17], p. 226). Recall that µ : B S d → [0 , + ∞ ) is said strictly positive if for every open set A ⊂ R n with A ∩ S d = ∅ , we have µ ( A ∩ S d ) > µ is defined by supp ( µ ) = { x ∈ S d ; µ ( N x ) > N x ∈ B S d of x } . Note that the strict positivity of µ , evidently implies the condition supp ( µ ) \ ∂S d = ∅ , which guarantees that ( C ) R S d B α ( t ) dµ ( t ) >
0, for all B α .Let us consider C + ( S d ) = { f : S d → R + ; f is continuous on S d } , endowedwith the norm k F k C ( S d ) = sup {| F ( x ) | ; x ∈ S d } .The first main result of this section consists in the following general quanti-tative estimates in pointwise and uniform approximation. Theorem 3.1.
For each fixed n ∈ N and x ∈ S d , let Γ n,x = { µ n,α,x } | α | = n be a family of bounded, monotone, submodular and strictly positive set functionson B S d .(i) For every f ∈ C + ( S d ) , x = ( x , ..., x d ) ∈ S d , n ∈ N , we have | M n, Γ n,x ( f )( x ) − f ( x ) | ≤ ω ( f ; M n, Γ n,x ( ϕ x )( x )) S d , where M n, Γ n,x ( f )( x ) is given by (2), k x k = qP di =1 x i , ϕ x ( t ) = k t − x k and ω ( f ; δ ) S d = sup {| f ( t ) − f ( x ) | ; t, x ∈ S d , k t − x k ≤ δ } .(ii) Suppose that the family Γ n,x does not depend on x . Then, for any f ∈ C + ( S d ) and n ∈ N , we get k M n, Γ n ( f ) − f k C ( S d ) ≤ K (cid:18) f ; ∆ n (cid:19) , where ∆ n = P di =1 k M n, Γ n ( | ϕ e i − x i | ) k C ( S d ) , K ( f ; t ) = inf g ∈ C ( S d ) {k f − g k C ( S d ) + t k∇ g k C ( S d ) } ,C ( S d ) is the subspace of all functions g ∈ C + ( S d ) with continuous partialderivatives ∂ i g , i ∈ { , ..., d } and k∇ g k C ( S d ) = max i = { ,...,d } {k ∂ i g k C ( S d ) } ,ϕ e i ( x ) = x i , i ∈ { , ..., d } , x = ( x , ..., x d ) , ( x ) = 1 , for all x ∈ S d . Proof. (i) For x ∈ S d , n ∈ N and | α | = n arbitrary fixed, let us consider T n,α,x : C + ( S d ) → R + defined by T n,α,x ( f ) = ( C ) Z S d f ( t ) P α ( t ) dµ n,α,x ( t ) , f ∈ C + ( S d ) . Based on the above Remark 2.3, (i), (ii), (iii) and reasoning exactly as in theproof of Lemma 3.1 in [10], we get | T n,α,x ( f ) − T n,α,x ( g ) | ≤ T n,α,x ( | f − g | ). Then,since T n,α,x is positively homogeneous, sublinear and monotonically increasing,we immediately get that M n, Γ n,x keeps the same properties and as a consequenceit follows | M n, Γ n,x ( f )( x ) − M n, Γ n,x ( g )( x ) | ≤ M n, Γ n,x ( | f − g | )( x ) , (3)5 n, Γ n,x ( λf ) = λM n, Γ n,x ( f ), M n, Γ n,x ( f + g ) ≤ M n, Γ n,x ( f ) + M n, Γ n,x ( g ) and that f ≤ g on S d implies M n, Γ n,x ( f ) ≤ M n, Γ n,x ( g ) on S d , for all λ ≥ f, g ∈ C + ( S d ), n ∈ N , | α | = n , x ∈ S d .Denoting e ( t ) = 1 for all t ∈ S d , since obviously M n, Γ n,x ( e )( x ) = 1 for all x ∈ S d and taking into account the properties in Remark 2.3, (i) and (3), forany fixed x we obtain | M n, Γ n,x ( f )( x ) − f ( x ) | = | M n, Γ n,x ( f ( t ))( x ) − M n, Γ n,x ( f ( x ))( x ) |≤ M n, Γ n,x ( | f ( t ) − f ( x ) | )( x ) . (4)But taking into account the properties of the modulus of continuity, for all t, x ∈ S d and δ >
0, we get | f ( t ) − f ( x ) | ≤ ω ( f ; k t − x k ) S d ≤ (cid:20) δ k t − x k + 1 (cid:21) ω ( f ; δ ) S d . (5)Now, from (4) and applying M n, Γ n,x to (5), by the properties of M n, Γ n,x men-tioned after the inequality (3), we immediately get | M n, Γ n,x ( f )( x ) − f ( x ) | ≤ (cid:20) δ M n, Γ n,x ( ϕ x )( x ) + 1 (cid:21) ω ( f ; δ ) S d . Choosing here δ = M n, Γ n,x ( ϕ x )( x ), we obtain the desired estimate.(ii) Let f, g ∈ C + ( S d ). We have f ( x ) − M n, Γ n ( f )( x )= f ( x ) − g ( x ) + M n, Γ n ( g )( x ) − M n, Γ n ( f )( x ) + g ( x ) − M n, Γ n ( g )( x ) , which, by using (3) too, implies | f ( x ) − M n, Γ n ( f )( x ) |≤ | f ( x ) − g ( x ) | + | M n, Γ n ( g )( x ) − M n, Γ n ( f )( x ) | + | g ( x ) − M n, Γ n ( g )( x ) |≤ | f ( x ) − g ( x ) | + M n, Γ n ( | g − f | )( x ) + | g ( x ) − M n, Γ n ( g )( x ) |≤ k f − g k C ( S d ) + | g ( x ) − M n, Γ n ( g )( x ) | . By following the lines in the proof of Theorem 4.5 in [4], since from the linesafter relation (3) in the above point (i), the operator M n, Γ n is monotone andsubadditive, for all g ∈ C ( S d ), x ∈ S d , we immediately get | g ( x ) − M n, Γ n ( g )( x ) |≤ M n, Γ n ( | g − g ( x ) | )( x ) ≤ k∇ g k C ( S d ) · M n, Γ n d X i =1 | ϕ e i − x i | ! ( x ) ≤ k∇ g k C ( S d ) · d X i =1 M n, Γ n ( | ϕ e i − x i | ) ( x ) ≤ k∇ g k C ( S d ) · ∆ n . k f − M n, Γ n ( f ) k C ( S d ) ≤ (cid:20) k f − g k C ( S d ) + ∆ n k∇ g k C ( S d ) (cid:21) , which immediately implies the required estimate in (ii). (cid:3) Remark 3.2.
The positivity of function f in Theorem 3.1, (i) and (ii), isnecessary because of the positive homogeneity of the Choquet integral used intheir proofs. However, if f is of arbitrary sign and lower bounded on S d with f ( x ) − m ≥
0, for all x ∈ S d , then the statement of Theorem 3.1, (i), (ii), canbe restated for the slightly modified Bernstein-Durrmeyer operator defined by M ∗ n, Γ n,x ( f )( x ) = M n, Γ n,x ( f − m )( x ) + m. Indeed, in the case of Theorem 3.1, (i), this is immediate from ω ( f − m ; δ ) S d = ω ( f ; δ ) S d and from M ∗ n, Γ n,x ( f )( x ) − f ( x ) = M n, Γ n,x ( f − m )( x ) − ( f ( x ) − m ).Note that in the case of Theorem 3.1, (ii), since we may consider here that m <
0, we immediately get the relations K ( f − m ; t ) = inf g ∈ C ( S d ) {k f − ( g + m ) k C ( S d ) + t k∇ g k C ( S d ) } = inf g ∈ C ( S d ) {k f − ( g + m ) k C ( S d ) + t k∇ ( g + m ) k C ( S d ) } = inf h ∈ C ( S d ) , h ≥ m {k f − h k C ( S d ) + t k∇ h k C ( S d ) } . In the particular case when the family Γ n,x does not depend on x and n , it isnatural to ask for quantitative estimates in the L -approximation of the Choquetintegrable functions (not necessarily continuous). If, for example, Γ n,x = { µ } , d = 1 and µ is a bounded, monotone and submodular set function, then for theBernstein-Durrmeyer-Choquet operators D n,µ ( f )( x ) = n X k =0 p n,k ( x ) · ( C ) R f ( t ) p n,k ( t ) dµ ( t )( C ) R p n,k ( t ) dµ ( t ) , p n,k ( x ) = (cid:18) nk (cid:19) x k (1 − x ) n − k , with f ∈ L µ meaning f is B [0 , measurable and k f k L µ = ( C ) R | f ( t ) | dµ ( t ) < ∞ , we get k D n,µ ( f ) k L µ ≤ n X k =0 ( C ) Z p n,k ( t ) | f ( t ) | dµ ( t ) ≤ n · k f k L µ , n ∈ N . This is due to the fact that ( C ) R f dµ is is not, in general, additive as functionof f (it is only subadditive).Therefore, quantitative estimates in L pµ -approximation by Bernstein-Durrmeyer-Choquet operators, remains, in the general case, an open question.However, in the particular case when the family of set functions Γ n,x reduces,for example, to two elements (one being a Choquet submodular set function µ δ ), for suitable defined Bernstein-Durrmeyer-Choquet operators, quantitative L pµ -approximation results, p ≥ L pµ = { f : [0 , → R ; f is B [0 , measurable and( C ) Z | f ( t ) | p dµ ( t ) < + ∞} ,L pµ, + = L pµ \ { f : [0 , → R + } ,K ( f ; t ) L pµ,δ = inf g ∈ C ([0 , {k f − g k L pµ + k f − g k L pδ + t k g ′ k C ([0 , } . It is easy to see that if µ ≤ δ then for all f ∈ L δ and t ≥
0, we have2 K ( f ; t/ L pµ ≤ K ( f ; t ) L pµ,δ ≤ K ( f ; t ) L pδ , where K is of usual form and withthe infimum taken for g ∈ C ([0 , p = 1, we have : Theorem 3.3.
Let µ be a bounded, monotone, submodular and strictlypositive set function on B [0 , and δ a bounded, strictly positive Borel measureon B [0 , , such that µ ( A ) ≤ δ ( A ) for all A ∈ B [0 , . Then, denoting L δ, + ⊂ L µ, + and defining the Bernstein-Durrmeyer-Choquet operators D n,δ,µ ( f )( x ) = n − X k =0 p n,k ( x ) · R f ( t ) p n,k ( t ) dδ ( t ) R p n,k ( t ) dδ ( t ) + x n · ( C ) R f ( t ) t n dµ ( t )( C ) R t n dµ ( t ) , for all f ∈ L δ, + , n ∈ N and denoting ϕ x ( t ) = | t − x | , we have k f − D n,δ,µ ( f ) k L µ ≤ K f ; k D n,δ,µ ( ϕ x ) k L µ ! L µ,δ . Proof.
Firstly, note that δ is monotone, submodular (in fact modular, i.e.submodular with equality) strictly positive and that for all f ∈ L δ, + we have R f ( t ) dδ ( t ) = ( C ) R f ( t ) dδ ( t ) (see Remark 2.3, (vii)). From here, from µ ≤ δ and from Definition 2.2, (ii), we immediately get ( C ) R f ( t ) dµ ( t ) ≤ R f ( t ) dδ ( t ),which means L δ, + ⊂ L µ, + and for all f ∈ L δ, + implies k D n,δ,µ ( f ) k L µ ≤ n − X k =0 ( C ) R p n,k ( x ) dµ ( x ) R p n,k ( t ) dδ ( t ) · Z f ( t ) p n,k ( t ) dδ +( C ) Z f ( t ) t n dµ ( t ) ≤ Z f ( t ) " n − X k =0 p n,k ( t ) dδ + ( C ) Z f ( t ) t n dµ ( t ) ≤ k f k L δ + k f k L µ . (6)Let f, g ∈ L δ, + . From | f ( t ) − D n,δ,µ ( f )( t ) |≤ | f ( t ) − g ( t ) | + | D n,δ,µ ( g )( t ) − D n,δ,µ ( f )( t ) | + | g ( t ) − D n,δ,µ ( g )( t ) | , µ , from the properties of the Choquet integral, ofthe operator D n,δ,µ (similar with those of M n, Γ n,x in the proof of Theorem 3.1,(i)) and from (6), we obtain k f − D n,δ,µ k L µ = ( C ) Z | f ( t ) − D n,δ,µ ( f )( t ) | dµ ( t ) ≤ ( C ) Z | f ( t ) − g ( t ) | dµ ( t ) + ( C ) Z | D n,δ,µ ( g )( t ) − D n,,δ,µ ( f )( t ) | dµ ( t )+( C ) Z | g ( t ) − D n,δ,µ ( g )( t ) | dµ ( t ) ≤ k f − g k L µ + ( C ) Z D n,δ,µ ( | g − f | )( t ) dµ ( t ) + k g − D n,δ,µ ( g ) k L µ ≤ k f − g k L µ + ( k f − g k L µ + k f − g k L δ ) + k g − D n,δ,µ ( g ) k L µ . It remains to estimate k g − D n,µ ( g ) k L µ . But from | g ( x ) − D n,µ ( g )( x ) | = | D n,µ ( g ( x ))( x ) − D n,µ ( g ( t ))( x ) | ≤ D n,µ ( | g ( x ) − g ( t ) | )( x )and since for g ∈ C ([0 , | g ( x ) − g ( t ) | ≤ k g ′ k C ([0 , | x − t | , applying D n,δ,µ , it follows D n,δ,µ ( | g ( x ) − g ( t ) | )( x ) ≤ k g ′ k C ([0 , · D n,µ ( ϕ x )( x ).Therefore, integrating above with respect to x and µ , we obtain k g − D n,δ,µ ( g ) k L µ ≤ k g ′ k C ([0 , · k D n,δ,µ ( ϕ x ) k L µ , which immediately leads to k f − D n,δ,µ ( f ) k L µ ≤ k f − g k L µ + k f − g k L δ + k g ′ k C ([0 , · k D n,δ,µ ( ϕ x ) k L µ ≤ k f − g k L µ + k f − g k L δ + k g ′ k C ([0 , · k D n,δ,µ ( ϕ x ) k L µ ! and to the conclusion of the theorem. (cid:3) In what follows, because of some difference with respect to the case p = 1,we extend separately Theorem 3.3 to the L pµ space with p > Theorem 3.4.
Let µ be a bounded, monotone, submodular, strictly positiveset function on B [0 , , which also is continuous by increasing sequences of sets,that is if A n ∈ B [0 , , n ∈ N , with A n ⊂ A n +1 , for all n and A := S ∞ n =1 A n ∈B [0 , , then lim n →∞ µ ( A n ) = µ ( A ) .Also, let δ be a bounded, strictly positive Borel measure on B [0 , , such that µ ( A ) ≤ δ ( A ) for all A ∈ B [0 , . Then, for any p > , L pδ, + ⊂ L pµ, + and for theBernstein-Durrmeyer-Choquet operators D n,δ,µ ( f )( x ) defined by Theorem 3.3,for all f ∈ L pδ, + = L pδ T { f : [0 , → [0 , + ∞ ) } and n ∈ N , we have k f − D n,δ,µ ( f ) k L pµ ≤ K f ; k D n,δ,µ ( ϕ x ) k L pµ ! L pµ,δ . roof. The proof for L pδ, + ⊂ L pµ, + follows exactly as in the proof of Theorem3.3.By the convexity of t p on [0 , + ∞ ), by P nk =0 p n,k ( x ) = 1, we easily arrive atthe inequalities (exactly as, for example, in the proof of Lemma 2.2 in [13]) k D n,δ,µ ( f ) k pL pµ ≤ ( C ) Z n − X k =1 p n,k ( x ) · (cid:16)R f ( t ) p n,k ( t ) dδ ( t ) (cid:17) p (cid:16)R p n,k ( t ) dδ ( t ) (cid:17) p + x n · (cid:16) ( C ) R f ( t ) t n dµ ( t ) (cid:17) p (cid:16) ( C ) R t n dµ ( t ) (cid:17) p dµ ( x ) . Applying the H¨older’s inequality for the integrals from the denominators (in thecase of Choquet integrals with respect to µ , the inequality is the same with thatfor the integrals with respect to the Borel measure δ , see. e.g., Theorem 3.5 in[16] or Theorem 2 in [5]) and reasoning as in the proof of Lemma 2.2 in [13] andas for formula (6) in the proof of Theorem 3.3, we easily arrive at k D n,δ,µ ( f ) k pL pµ ≤ k f k L pµ + k f k L pδ , for all f ∈ L pδ, + . Then, since H¨older’s inequality for the Choquet integral with respect to µ impliesthe Minkowski inequality (see, e.g., Theorem 3.7 in [16] or Theorem 2 in [5]),using the above inequality and exactly the reasonings in the proof of Theorem2.1 in [13], we arrive at the desired inequality in the statement.It remains to discuss the requirement on µ to be continuous by increasingsequences of sets. This is due to the fact that for Choquet integrals, the H¨older’sinequality hold only if both integrals from its right-hand side are not equal tozero (see the proofs of Theorem 3.5 in [16] or of Theorem 2 in [5]).To have valid the H¨older’s inequality in its full generality, we need that for F ≥
0, ( C ) R F ( t ) dµ = 0 if and only if F ( t ) = 0, µ almost everywhere on [0 , µ is continuous by increasingsequences of sets, then the above mentioned property holds. (cid:3) Remark 3.5.
Concrete choices for µ and δ in Theorem 3.3 can be, forexample, δ ( A ) = m ( A ) and µ ( A ) = sin[ m ( A )], where m is the Lebesgue measureon B [0 , . Indeed, µ ( A ) ≤ m ( A ) for all A ∈ B [0 , and since sin is concave on[0 ,
1] and sin (0) = 0, by Remark 2.3, (vi) it follows that µ is bounded, monotoneand submodular. Remark 3.6.
It is easy to see that another Bernstein-Durrmeyer-Choquetoperator satisfying the estimates in Theorems 3.3 and 3.4, can be defined by˜ D n,δ,µ ( f )( x ) = (1 − x ) n ( C ) R f ( t )(1 − t ) n dµ ( t )( C ) R (1 − t ) n dµ ( t ) + n X k =1 p n,k ( x ) R f ( t ) p n,k ( t ) dδ ( t ) R p n,k ( t ) dδ ( t ) . Also, defining D ⋆n,δ,µ ( f )( x ) = (1 − x ) n ( C ) R f ( t )(1 − t ) n dµ ( t )( C ) R (1 − t ) n dµ ( t ) + n − X k =1 p n,k ( x ) R f ( t ) p n,k ( t ) dδ ( t ) R p n,k ( t ) dδ ( t )10 x n · ( C ) R f ( t ) t n dµ ( t )( C ) R t n dµ ( t ) , by similar reasonings with those in the proofs of Theorems 3.3 and 3.4, for any p ≥ k f − D ⋆n,δ,µ ( f ) k L pµ ≤ K (cid:18) f ; k D ⋆n,δ,µ ( ϕ x ) k L pµ (cid:19) . Remark 3.7.
For δ a bounded Borel measure on B [0 , , denote by D n,δ theclassical Bernstein-Durrmeyer operator (i.e. with all the integrals in terms of δ ). By Theorem 4.5 in [4], we have the estimate k D n,δ ( f ) − f k L δ ≤ K f ; k D n,δ ( ϕ x ) k L δ ! L δ , for all f ∈ L δ , where K ( f ; t ) L δ = inf g ∈ C ([0 , {k f − g k L δ + t k g ′ k C ([0 , } .Comparing with the estimate for k D n,δ,µ ( f ) − f k L µ in Theorem 3.3 and tak-ing into account that k f − g k L µ ≤ k f − g k L δ and k D n,δ ( ϕ x ) k L µ ≤ k D n,δ ( ϕ x ) k L δ ,it follows that it is possible that in some cases, D n,δ,µ ( f ) in Theorem 3.3, ap-proximates better f ∈ L δ, + in the L µ -”norm” than approximate D n,δ ( f ) thesame function f but in the L δ -norm. Since the estimates in Theorem 3.1 are of very general and abstract form, involv-ing the apparently difficult to be calculated Choquet integrals, it is of interestto obtain in some particular cases, concrete expressions for the order of approx-imation.In this sense, we will apply Theorem 3.1, (i), for d = 1 and for some specialchoices of the submodular set functions.Thus, we will consider the case of the measures of possibility. Denot-ing p n,k ( x ) = (cid:0) nk (cid:1) x k (1 − x ) n − k , let us define λ n,k ( t ) = p n,k ( t ) k k n − n ( n − k ) n − k ( nk ) = t k (1 − t ) n − k k k n − n ( n − k ) n − k , k = 0 , ..., n . Here, by convention we consider 0 = 1, so that thecases k = 0 and k = n have sense.By considering the root kn of p ′ n,k ( x ), it is easy to see that max { p n,k ( t ); t ∈ [0 , } = k k n − n ( n − k ) n − k (cid:0) nk (cid:1) , which implies that each λ n,k is a possibilitydistribution on [0 , P λ n,k the possibility measure induced by λ n,k and Γ n,x := Γ n = { P λ n,k } nk =0 (i.e. Γ is independent of x ), the nonlinearBernstein-Durrmeyer polynomial operators given by (2), in terms of the Choquetintegrals with respect to the set functions in Γ n , will become D n, Γ n ( f )( x ) = n X k =0 p n,k ( x ) · ( C ) R f ( t ) t k (1 − t ) n − k dP λ n,k ( t )( C ) R t k (1 − t ) n − k dP λ n,k ( t ) . (7)11t is easy to see that any possibility measure P λ n,k is bounded, monotone, sub-modular and strictly positive, n ∈ N , k = 0 , , ..., n , so that we are under thehypothesis of Theorem 3.1, (i).We can state the following result. Theorem 4.1. If D n, Γ n ( f )( x ) is given by (7), then for every f ∈ C + ([0 , , x ∈ [0 , and n ∈ N , n ≥ , we have | D n, Γ n ( f )( x ) − f ( x ) | ≤ ω f ; (1 + √ p x (1 − x ) + √ √ x √ n + 1 n ! [0 , . For its proof, we need the following auxiliary result.
Lemma 4.2.
Let n ∈ N , n ≥ and x ∈ [0 , . Denoting A n,k ( x ) := sup {| t − x | t k (1 − t ) n − k ; t ∈ [0 , } =max { sup { ( t − x ) t k (1 − t ) n − k ; t ∈ [ x, } , sup { ( x − t ) t k (1 − t ) n − k ; t ∈ [0 , x ] }} , with the convention = 1 , for all k = 0 , ..., n we have A n,k ( x ) = max { ( t − x ) t k (1 − t ) n − k , ( x − t ) t k (1 − t ) n − k } , with t , t given by t = nx + k + 1 − √ ∆2( n + 1) , t = nx + k + 1 + √ ∆2( n + 1) , (8) where ∆ = ( nx + k + 1) − kx ( n + 1) = n (cid:20) ( x + ( k + 1) /n ) − x kn · n + 1 n (cid:21) = ( nx − k ) + 2 x ( n − k ) + 2 k (1 − x ) + 1 ≥ . Proof.
Let us denote H n,k ( t ) = t k (1 − t ) n − k | t − x | , with k ∈ { , ..., n } . Wehave two cases : (i) 1 ≤ k ≤ n − k = 0 or k = n .Case (i). For t ∈ [ x,
1] we obtain H n,k ( t ) = ( t − x ) t k (1 − t ) n − k and from H ′ n,k ( t ) = t k − (1 − t ) n − k − [ − t ( n + 1) + t ( nx + k + 1) − kx ] = 0, it follows − t ( n + 1) + t ( nx + k + 1) − kx = 0, which has the discriminant∆ = ( nx + k + 1) − kx ( n + 1) = ( nx − k ) + 2 x ( n − k ) + 2 k (1 − x ) + 1 ≥ . Therefore, the quadratic equation has two real distinct solutions t < t t = nx + k + 1 − √ ∆2( n + 1) , t = nx + k + 1 + √ ∆2( n + 1) , where by simple calculation we derive 0 ≤ t < t ≤
1. Also, since H n,k (0) = H n,k ( x ) = H n,k (1) = 0 and H n,k ( t ) ≥ t ∈ [ x, ≤ t ≤ x ≤ t ≤
1, with t = t maximumpoint on [ x,
1] for H n,k ( t ). 12imilarly, for t ∈ [0 , x ], since H n,k ( t ) = ( x − t ) t k (1 − t ) n − k , using the abovereasonings we obtain H ′ n,k ( t ) = t k − (1 − t ) n − k − [ t ( n + 1) − t ( nx + k + 1) + kx ]and that t is a maximum point of H n,k ( t ) on [0 , x ].In conclusion, with t , t given by (8), we get A n,k ( x ) = max { ( t − x ) t k (1 − t ) n − k , ( x − t ) t k (1 − t ) n − k } . Case (ii). Suppose first that k = 0. By the calculation from the case (i),for t ∈ [ x,
1] we get 0 = t ≤ x ≤ t = nx +1 n +1 ≤ H n, ( t ) ≥ H n, ( x ) = H n, (1) = 0, which by similar graphical reasonings leads to the fact that themaximum of H n, ( t ) on [ x,
1] is H n, ( t ) = ( t − x )(1 − t ) n . Therefore, werecapture the case (i) with the convention that 0 = 1. Similarly, for t ∈ [0 , x ],we get that the maximum of H n, ( t ) is H n, ( t ) = ( x − t )(1 − t ) n The subcase k = n is similar, which proves the lemma. (cid:3) Proof of Theorem 4.1.
According to Theorem 3.1, (i), we have to estimate D n, Γ n ( ϕ x )( x ) = n X k =0 p n,k ( x ) · ( C ) R | t − x | t k (1 − t ) n − k dP λ n,k ( t )( C ) R t k (1 − t ) n − k dP λ n,k ( t ) . First of all, by Definition 2.2, (ii), we get( C ) Z t k (1 − t ) n − k dP λ n,k ( t ) = Z + ∞ P λ n,k ( { t ∈ [0 , t k (1 − t ) n − k ≥ β } ) dβ = Z P λ n,k ( { t ∈ [0 , t k (1 − t ) n − k ≥ β } ) dβ = Z sup { λ n,k ( s ); s ∈ { t ∈ [0 , t k (1 − t ) n − k ≥ β }} dβ = 1 k k n − n ( n − k ) n − k · Z sup { s k (1 − s ) n − k ; s ∈ { t ∈ [0 , t k (1 − t ) n − k ≥ β }} dβ. For simplicity, denote E n,k = k k n − n ( n − k ) n − k , where again we take 0 = 1.Since for β > E n,k we have { t ∈ [0 , t k (1 − t ) n − k ≥ β } = ∅ and since we cantake sup { s k (1 − s ) n − k ; s ∈ ∅} = 0, it follows( C ) Z t k (1 − t ) n − k dP λ n,k ( t )= 1 E n,k · Z E n,k sup { s k (1 − s ) n − k ; s ∈ { t ∈ [0 , t k (1 − t ) n − k ≥ β }} dβ = 1 E n,k · Z E n,k E n,k dβ = E n,k . (9)13n the other hand, denoting A n,k ( x ) = sup {| t − x | t k (1 − t ) n − k ; t ∈ [0 , } , byRemark 2.3, (iii), (v) and by Lemma 4.2, for t < t in (8) we obtain( C ) Z | t − x | t k (1 − t ) n − k dP λ n,k ( t ) ≤ ( C ) Z A n,k ( x ) dP λ n,k ( t )= A n,k ( x )( C ) Z dP λ n,k ( t ) = max { ( t − x ) t k (1 − t ) n − k , ( x − t ) t k (1 − t ) n − k }≤ ( t − x ) t k (1 − t ) n − k + ( x − t ) t k (1 − t ) n − k . Since t k (1 − t ) n − k k k n − n ( n − k ) n − k ≤ t k (1 − t ) n − k k k n − n ( n − k ) n − k ≤ A n,k ( x ) k k n − n ( n − k ) n − k ≤ ( t − x ) · t k (1 − t ) n − k k k n − n ( n − k ) n − k + ( x − t ) · t k (1 − t ) n − k k k n − n ( n − k ) n − k ≤ t − t = √ ∆ n + 1 ≤ p ( nx − k ) + 2 x ( n − k ) + 2 k (1 − x ) + 1 n ≤ p ( x − k/n ) + 2 x/n + (2 k/n ) · (1 − x ) /n + 1 /n ≤ | x − k/n | + √ x/ √ n + ( √ k/ √ n ) · p (1 − x ) /n + 1 /n, this immediately implies D n, Γ n ( ϕ x )( x ) ≤ n X k =0 p n,k ( x )( | x − k/n | + √ x/ √ n + p k/n · p (1 − x ) /n + 1 /n ) ≤ p x (1 − x ) √ n + √ x √ n + √ p x (1 − x ) √ n + 1 n = (1 + √ p x (1 − x ) + √ √ x √ n + 1 n . Above we have used the well-known estimate P nk =0 p n,k ( x ) | x − k/n | ≤ √ x (1 − x ) √ n and the Cauchy-Schwarz inequality for Bernstein polynomials, B n ( f )( x ) ≤ p B n ( f )( x ), applied for f ( t ) = √ t .Finally, applying Theorem 3.1, (i), the proof of Theorem 4.1 follows. (cid:3) Remark 4.3.
For µ = √ m with m denoting the Lebesgue measure on[0 , D n,µ ( f )( x ) = n X k =0 p n,k ( x ) · ( C ) R f ( t ) t k (1 − t ) n − k dµ ( C ) R t k (1 − t ) n − k dµ . It is worth noting that the uniform convergence of this D n,µ ( f ) to f , followsdirectly from the general Theorem 3.2 in [10]. Also, it can be obtained by usingthe nonlinear Feller kind scheme expressed by Theorem 3.1 in [9] (combinedwith Remark 3.2 there), since by direct calculation we can show that D n,µ ( e )converges uniformly to e and D n,µ (( · − x ) ) converges uniformly to 0, on [0 , emark 4.4. Since the Bernstein-Durrmeyer-Choquet operators in thispaper can be defined with respect to a family of Borel or Choquet measures,combined in various ways, this fact offers a very high flexibility and generality,allowing to construct operators having even better approximation properties. Afirst example for this flexibility is shown by Theorem 3.4 and Remark 3.6.For the second example, let us replace in formula (7) the family Γ n of mea-sures of possibilities P λ n,k , k = 0 , ..., n , by the family consisting in the Diracmeasures δ k/n , k = 0 , , ..., n −
1, (which are Borel measures and therefore withthe corresponding Choquet integrals reducing to the classical ones) togetherwith a monotone, submodular, strictly positive set function µ . Then, denotingby B n ( f )( x ) the classical Bernstein operators, for D n, Γ n in (7) we get D n, Γ n ( f )( x ) − f ( x ) = " n − X k =0 p n,k ( x ) f (cid:18) kn (cid:19) + x n · ( C ) R f ( t ) t n dµ ( t )( C ) R t n dµ ( t ) − f ( x )= B n ( f )( x ) − f ( x ) + x n " ( C ) R f ( t ) t n dµ ( t )( C ) R t n dµ ( t ) − f (1) . Suppose now that f ≥ ,
1] and,for example, that µ ( A ) = p m ( A ) or µ ( A ) = sin[ m ( A )], with m the Lebesguemeasure. The strict convexity implies B n ( f )( x ) − f ( x ) > x ∈ (0 ,
1) andthe property of f to be strictly increasing easily implies( C ) R f ( t ) t n dµ ( t )( C ) R t n dµ ( t ) − f (1) < f (1) · ( C ) R t n dµ ( t )( C ) R t n dµ ( t ) − f (1) = 0 . Therefore, in this case we get | D n, Γ n ( f )( x ) − f ( x ) | < max ( | B n ( f )( x ) − f ( x ) | , x n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( C ) R f ( t ) t n dµ ( t )( C ) R t n dµ ( t ) − f (1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)) , i.e. for x ∈ (0 , D n, Γ n ( f )( x ) approximates better than B n ( f )( x ).Here it is clear that B n ( f )( x ) can also be viewed as the Bernstein-Durrmeyeroperators in the case when Γ n is composed by the Dirac measures δ k/n , k =0 , ..., n , where we note that although the Dirac measures are not strictly positive,however the Bernstein-Durrmeyer operators attached to them are well defined.This fact contrasts with the classical case in [1] when Γ n is composed by only oneset function, independent of n , and when the strict positivity of the set functionis necessarily for the convergence (see Theorem 1 in [1]). In other words, thestrict positivity of the set functions in Theorem 3.1 is not always necessary.For another example, let us consider the genuine Bernstein-Durrmeyer-Cho-quet operators given by U n, Γ n ( f )( x ) = p n, ( x ) · ( C ) R f ( t )(1 − t ) n dν n, ( C ) R (1 − t ) n dν n, + p n,n ( x ) · ( C ) R f ( t ) t n dν n,n ( C ) R t n dν n,n n − X k =1 p n,k ( x ) · ( C ) R f ( t ) p n − ,k − ( t ) dµ n − ,k − ( t )( C ) R p n − ,k − ( t ) dµ n − ,k − ( t ) , where Γ n = { ν n, , ν n,n , µ n − ,k − , k = 1 , ..., n − } .Let us denote by G n ( f )( x ), the classical genuine Bernstein-Durmeyer oper-ator (see, e.g., [11]). Choosing in Γ n the set functions µ n − ,k − , k = 1 , ..., n − ν n, = δ (as Dirac measure) and ν n,n as a monotone,submodular and strictly positive set function, we immediately obtain U n, Γ n ( f )( x ) − f ( x ) = G n ( f )( x ) − f ( x ) + x n " ( C ) R f ( t ) t n dν n,n ( t )( C ) R t n dν n,n ( t ) − f (1) . Since the strict convexity of f implies G n ( f )( x ) − f ( x ) > x ∈ (0 , f is strictly convex and strictly increasing on [0 ,
1] (and,for example, ν n,n ( A ) = p m ( A ) or ν n,n ( A ) = sin[ m ( A )]), then U n, Γ n ( f )( x )approximates better f on on (0 ,
1) than the classical genuine operator, G n ( f )( x ). Remark 4.5.
Recall that in [9], Example 4.2, for the nonlinear Picard-Choquet operators we have obtained a general estimate similar to that for theclassical Picard operators, while for particular functions of the form f ( x ) = M e − Ax , M, A >
0, we got there essentially better error estimates.
Remark 4.6.
In [13] applications of the classical Bernstein-Durrmeyer op-erators to learning theory are presented. Taking onto account the very recentapplications of the Choquet integral to learning theory (see, e.g., [14] and thereferences therein), it becomes of interest to see for potential applications ofthe Bernstein-Durrmeyer-Choquet operators to learning theory. Also, takinginto account the applications of the classical Bernstein-Durrmeyer operators inregression estimation in, e.g., [15] and the very recent applications of the Cho-quet integral to regression model, see, e.g., [12], it would be of interest to seefor possible applications of the Bernstein-Durrmeyer-Choquet operators to theregression model.
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