Positive L p -bounded Dunkl-type generalized translation operator and its applications
aa r X i v : . [ m a t h . C A ] D ec POSITIVE L p -BOUNDED DUNKL-TYPE GENERALIZEDTRANSLATION OPERATOR AND ITS APPLICATIONS D. V. GORBACHEV, V. I. IVANOV, AND S. YU. TIKHONOV
Abstract.
We prove that the spherical mean value of the Dunkl-type gener-alized translation operator τ y is a positive L p -bounded generalized translationoperator T t . As application, we prove the Young inequality for a convolutiondefined by T t , the L p -boundedness of τ y on a radial functions for p > , the L p -boundedness of the Riesz potential for the Dunkl transform and direct andinverse theorems of approximation theory in L p -spaces with the Dunkl weight. Contents
1. Introduction 12. Notation 23. Generalized translation operators and convolutions 54. Boundedness of the Riesz potential 135. Entire functions of exponential type and Plancherel–Polya–Boas-typeinequalities 156. Jackson’s inequality and equivalence of modulus of smoothness and K -functional 226.1. Smoothness characteristics and K -functional 226.2. Main results 276.3. Properties of the de la Vall´ee Poussin type operators 286.4. Proofs of Theorem 6.6 and 6.8 337. Some inequalities for entire functions 348. Realization of K -functionals and moduli of smoothness 389. Inverse theorems of approximation theory 40References 421. Introduction
During the last three decades, many important elements of harmonic analysiswith Dunkl weight on R d and S d − were proved; see, e.g., the papers by C.F. Dunkl[14, 15, 16], M. R¨osler [40, 41, 42, 43], M.F.E. de Jeu [24, 25], K. Trim`eche [52, 53],Y. Xu [54, 55], and the recent works [1, 11, 12, 19, 20]. Date : December 5, 2018.1991
Mathematics Subject Classification.
Key words and phrases.
Dunkl transform, generalized translation operator, convolution, Rieszpotential.The work of D.V. Gorbachev and V.I. Ivanov is supported by the Russian Science Foundationunder grant N 18-11-00199 and performed in Tula State University. The work of S.Yu. Tikhonovis partially supported by MTM 2017-87409-P, 2017 SGR 358, and by the CERCA Programmeof the Generalitat de Catalunya.
Yet there are still several gaps in our knowledge of Dunkl harmonic analysis. Inparticular, Young’s convolution inequality, several important polynomial inequal-ities, and basic approximation estimates are not established in the general case.One of the main reasons is the lack of tools related to the translation operator.Needless to say, the standard translation operator f f ( · + y ) plays a crucial roleboth in classical approximation theory and harmonic analysis, in particular, to in-troduce several smoothness characteristics of f . In Dunkl analysis, its analogue isthe generalized translation operator τ y defined by M. R¨osler [40]. Unfortunately,the L p -boundedness of τ y is not obtained in general.To overcome this difficulty, the spherical mean value of the translation operator τ y was introduced in [28] and it was studied in [42], where, in particular, itspositivity was shown. Our main goal in this paper is to prove that this operatoris a positive L p -bounded operator T t , which may be considered as a generalizedtranslation operator. It is worth mentioning that this operator can be applied toproblems where it is essential to deal with radial multipliers. This is because byvirtue of T t we can define the convolution operator which coincides with the knownconvolution introduced by S. Thangavelu and Y. Xu in [48] using the operator τ y .For this convolution we prove the Young inequality and, subsequently, an L p -boundedness of the operator τ y on a radial functions for p > . For ≤ p ≤ itwas proved in [48].Let us mention here two applications of the operator T t . The first one is theRiesz potential defined in [49], where its boundedness properties were obtained forthe reflection group Z d . For the general case see [21]. Using the L p -boundednessof the operator T t allows us to give a different simple proof, which follows ideasof [49]. Another application is basic inequalities of approximation theory in theweighted L p spaces. With the help of the operator T t one can define moduli ofsmoothness, which are equivalent to the K -functionals, and prove the direct andinverse approximation theorems. For the reflection group Z d , basic approximationinequalities were studied in [11, 12].The paper is organized as follows. In the next section, we give some basic no-tation and facts of Dunkl harmonic analysis. In Section 3, we study the operator T t , define a convolution operator and prove the Young inequality. As a conse-quence, we obtain an L p -boundedness of the operator τ y on a radial functions.The weighted Riesz potential is studied in Section 4. Section 5 consists of a studyof interrelation between several classes of entire functions. We also obtain mul-tidimensional weighted analogues of Plancherel–Polya–Boas inequalities, whichare of their own interest. In Section 6 we introduce moduli of smoothness andthe K -functional, associated to the Dunkl weight, and prove equivalence betweenthem as well as the Jackson inequality. Section 7 consists of weighted analoguesof Nikol’skiˇi, Bernstein, and Boas inequalities for entire functions of exponentialtype. In Section 8, we obtain that moduli of smoothness are equivalent to therealization of the K -functional. We conclude with Section 9, where we prove theinverse theorems in L p -spaces with the Dunkl weight.2. Notation
In this section, we recall the basic notation and results of Dunkl harmonicanalysis, see, e.g., [43]. p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 3 Throughout the paper, h x, y i denotes the standard Euclidean scalar product in d -dimensional Euclidean space R d , d ∈ N , equipped with a norm | x | = p h x, x i .For r > we write B r = { x ∈ R d : | x | ≤ r } . Define the following function spaces: · C ( R d ) the space of continuous functions, · C b ( R d ) the space of bounded continuous functions with the norm k f k ∞ =sup R d | f | , · C ( R d ) the space of continuous functions which vanish at infinity, · C ∞ ( R d ) the space of infinitely differentiable functions, · C ∞ Π ( R d ) the space of infinitely differentiable functions whose derivativeshave polynomial growth at infinity, · S ( R d ) the Schwartz space, · S ′ ( R d ) the space of tempered distributions, · X ( R + ) the space of even functions from X ( R ) , where X is one of the spacesabove, · X rad ( R d ) the subspace of X ( R d ) consisting of radial functions f ( x ) = f ( | x | ) .Let a finite subset R ⊂ R d \ { } be a root system, R + a positive subsystem of R , G ( R ) ⊂ O ( d ) the finite reflection group, generated by reflections { σ a : a ∈ R } ,where σ a is a reflection with respect to hyperplane h a, x i = 0 , k : R → R + a G -invariant multiplicity function. Recall that a finite subset R ⊂ R d \ { } is calleda root system, if R ∩ R a = { a, − a } and σ a R = R for all a ∈ R. Let v k ( x ) = Y a ∈ R + |h a, x i| k ( a ) be the Dunkl weight, c − k = Z R d e −| x | / v k ( x ) dx, dµ k ( x ) = c k v k ( x ) dx, and L p ( R d , dµ k ) , < p < ∞ , be the space of complex-valued Lebesgue measurablefunctions f for which k f k p,dµ k = (cid:16)Z R d | f | p dµ k (cid:17) /p < ∞ . We also assume that L ∞ ≡ C b and k f k ∞ ,dµ k = k f k ∞ . Example.
If the root system R is {± e , . . . , ± e d } , where { e , . . . , e d } is an or-thonormal basis of R d , then v k ( x ) = Q dj =1 | x j | k j , k j ≥ , G = Z d .Let D j f ( x ) = ∂f ( x ) ∂x j + X a ∈ R + k ( a ) h a, e j i f ( x ) − f ( σ a x ) h a, x i , j = 1 , . . . , d be differential-differences Dunkl operators and ∆ k = P dj =1 D j be the Dunkl Lapla-cian. The Dunkl kernel e k ( x, y ) = E k ( x, iy ) is a unique solution of the system D j f ( x ) = iy j f ( x ) , j = 1 , . . . , d, f (0) = 1 , D. V. GORBACHEV, V. I. IVANOV, AND S. YU. TIKHONOV and it plays the role of a generalized exponential function. Its properties are similarto those of the classical exponential function e i h x,y i . Several basic properties followfrom an integral representation [41]: e k ( x, y ) = Z R d e i h ξ,y i dµ kx ( ξ ) , where µ kx is a probability Borel measure, whose support is contained in co( { gx : g ∈ G ( R ) } ) , the convex hull of the G -orbit of x in R d . In particular, | e k ( x, y ) | ≤ .For f ∈ L ( R d , dµ k ) , the Dunkl transform is defined by the equality F k ( f )( y ) = Z R d f ( x ) e k ( x, y ) dµ k ( x ) . For k ≡ , F is the classical Fourier transform F . We also note that F k ( e −| · | / )( y ) = e −| y | / and F − k ( f )( x ) = F k ( f )( − x ) . Let(2.1) A k = n f ∈ L ( R d , dµ k ) ∩ C ( R d ) : F k ( f ) ∈ L ( R d , dµ k ) o . Let us now list several basic properties of the Dunkl transform.
Proposition 2.1. (1)
For f ∈ L ( R d , dµ k ) , F k ( f ) ∈ C ( R d ) . (2) If f ∈ A k , we have the pointwise inversion formula f ( x ) = Z R d F k ( f )( y ) e k ( x, y ) dµ k ( y ) . (3) The Dunkl transform leaves the Schwartz space S ( R d ) invariant. (4) The Dunkl transform extends to a unitary operator in L ( R d , dµ k ) . Let λ ≥ − / and J λ ( t ) be the classical Bessel function of degree λ and j λ ( t ) = 2 λ Γ( λ + 1) t − λ J λ ( t ) be the normalized Bessel function. Set b − λ = Z ∞ e − t / t λ +1 dt = 2 λ Γ( λ + 1) , dν λ ( t ) = b λ t λ +1 dt, t ∈ R + . The norm in L p ( R + , dν λ ) , ≤ p < ∞ , is given by k f k p,dν λ = (cid:16)Z R + | f ( t ) | p dν λ ( t ) (cid:17) /p . Define k f k ∞ = ess sup t ∈ R + | f ( t ) | .The Hankel transform is defined as follows H λ ( f )( r ) = Z R + f ( t ) j λ ( rt ) dν λ ( t ) , r ∈ R + . It is a unitary operator in L ( R + , dν λ ) and H − λ = H λ [2, Chap. 7].Note that if λ = d/ − , the Hankel transform is a restriction of the Fouriertransform on radial functions and if λ = λ k = d/ − P a ∈ R + k ( a ) of the Dunkltransform.Let S d − = { x ′ ∈ R d : | x ′ | = 1 } be the Euclidean sphere and dσ k ( x ′ ) = a k v k ( x ′ ) dx ′ be the probability measure on S d − . We have(2.2) Z R d f ( x ) dµ k ( x ) = Z ∞ Z S d − f ( tx ′ ) dσ k ( x ′ ) dν λ k ( t ) . p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 5 We need the following partial case of the Funk–Hecke formula [55](2.3) Z S d − e k ( x, ty ′ ) dσ k ( y ′ ) = j λ k ( t | x | ) . Throughout the paper, we will assume that A . B means that A ≤ CB with aconstant C depending only on nonessential parameters.3. Generalized translation operators and convolutions
Let y ∈ R d be given. M. R¨osler [40] defined a generalized translation operator τ y in L ( R d , dµ k ) by the equation F k ( τ y f )( z ) = e k ( y, z ) F k ( f )( z ) . Since | e k ( y, z ) | ≤ then k τ y k → ≤ . If f ∈ A k (recall that A k is given by (2.1)),then, for any x, y ∈ R d ,(3.1) τ y f ( x ) = Z R d e k ( y, z ) e k ( x, z ) F k ( f )( z ) dµ k ( z ) . Note that S ( R d ) ⊂ A k ⊂ L ( R d , dµ k ) . K. Trim`eche [53] extended the operator τ y on C ∞ ( R d ) .The explicit expression of τ y f is known only in the case of the reflection group Z d . In particular, in this case τ y f is not a positive operator [39]. Note that in thecase of symmetric group S d the operator τ y f is also not positive [48].It remains an open question whether τ y f is an L p bounded operator on S ( R d ) for p = 2 . It is known ([39, 48]) only for G = Z d . Note that a positive answerwould follow from the L -boundedness.Let λ k = d/ − X a ∈ R + k ( a ) . We have λ k ≥ − / and, moreover, λ k = − / only if d = 1 and k ≡ . In whatfollows we assume that λ k > − / .Define another generalized translation operator T t : L ( R d , dµ k ) → L ( R d , dµ k ) , t ∈ R , by the relation F k ( T t f )( y ) = j λ k ( t | y | ) F k ( f )( y ) . Since | j λ k ( t ) | ≤ , it is a bounded operator such that k T t k → ≤ and T t f ( x ) = Z R d j λ k ( t | y | ) e k ( x, y ) F k ( f )( y ) dµ k ( y ) . This gives T t = T − t . If f ∈ A k , then from (2.3) and (3.1) we have (pointwise)(3.2) T t f ( x ) = Z R d j λ k ( t | y | ) e k ( x, y ) F k ( f )( y ) dµ k ( y ) = Z S d − τ ty ′ f ( x ) dσ k ( y ′ ) . D. V. GORBACHEV, V. I. IVANOV, AND S. YU. TIKHONOV
Note that the operator T t is self-adjoint. Indeed, if f, g ∈ A k , then Z R d T t f ( x ) g ( x ) dµ k ( x ) = Z R d Z R d j λ k ( t | y | ) e k ( x, y ) F k ( f )( y ) dµ k ( y ) g ( x ) dµ k ( x )= Z R d j λ k ( t | y | ) F k ( f )( y ) F k ( g )( − y ) dµ k ( y )= Z R d j λ k ( t | y | ) F k ( g )( y ) F k ( f )( − y ) dµ k ( y )= Z R d f ( x ) T t g ( x ) dµ k ( x ) . M. R¨osler [42] proved that the spherical mean (with respect to the Dunkl weight)of the operator τ y , i.e., R S d − τ ty ′ f ( x ) dσ k ( y ′ ) , is a positive operator on C ∞ ( R d ) andobtained its integral representation. This implies that T t is a positive operator on C ∞ ( R d ) and, moreover, for any t ∈ R , x ∈ R d ,(3.3) T t f ( x ) = Z R d f ( z ) dσ kx,t ( z ) , where σ kx,t is a probability Borel measure,(3.4) supp σ kx,t ⊂ [ g ∈ G { z ∈ R d : | z − gx | ≤ t } and the mapping ( x, t ) → σ kx,t is continuous with respect to the weak topology onprobability measures.The representation (3.3) gives a natural extension of the operator T t on C b ( R d ) ,namely, for f ∈ C b ( R d ) we define T t f ( x ) ∈ C ( R × R d ) by (3.3) and, moreover, theestimate k T t f k ∞ ≤ k f k ∞ holds.Note that for k ≡ , T t is the usual spherical mean(3.5) T t f ( x ) = S t f ( x ) = Z S d − f ( x + ty ′ ) dσ ( y ′ ) . Theorem 3.1. If ≤ p ≤ ∞ , then, for any t ∈ R and f ∈ S ( R d ) , (3.6) k T t f k p,dµ k ≤ k f k p,dµ k . Remark . (i) The inequality k T t f k p,dµ k ≤ c k f k p,dµ k was proved in [48] for G = Z d .(ii) Since S ( R d ) is dense in L p ( R d , dµ k ) , ≤ p < ∞ , then for any t ∈ R + theoperator T t can be defined on L p ( R d , dµ k ) and estimate (3.6) holds.(iii) If d = 1 , v k ( x ) = | x | λ +1 , λ > − / , inequality (3.6) was proved in [7]. Inthis case the integral representation of T t is of the form: T t f ( x ) = c λ Z π { f ( A )(1 + B ) + f ( − A )(1 − B ) } sin λ ϕ dϕ, where, for ( x, t ) = (0 , ,(3.7) c λ = Γ( λ + 1) √ π Γ( λ + 1 / , A = p x + t − xt cos ϕ, B = x − t cos ϕA . If λ = − / , i.e., k ≡ , then T t f ( x ) = (cid:0) f ( x + t ) + f ( x − t ) (cid:1) . p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 7 Proof.
Let t ∈ R + be given and the operator T t be defined on S ( R d ) by (3.3).Using (3.2), we have sup {k T t f k : f ∈ S ( R d ) , k f k ≤ } ≤ and T t can be extended to the space L ( R d , dµ k ) with preservation of norm, more-over, this extension coincides with (3.2). Moreover, (3.3) yields(3.8) sup {k T t f k ∞ : f ∈ S ( R d ) , k f k ∞ ≤ } ≤ . Since the operator T t is self-adjoint, then by (3.8) sup {k T t f k ,dµ k : f ∈ S ( R d ) , k f k ,dµ k ≤ } = sup nZ R d T t f g dµ k : f, g ∈ S ( R d ) , k f k ,dµ k ≤ , k g k ∞ ≤ o = sup nZ R d f T t g dµ k : f, g ∈ S ( R d ) , k f k ,dµ k ≤ , k g k ∞ ≤ o = sup {k T t g k ∞ : g ∈ S ( R d ) , k g k ∞ ≤ } ≤ . Hence, T t can be extended to L ( R d , dµ k ) with preservation of the norm such thatthis extension coincides with (3.2) on L ( R d , dµ k ) ∩ L ( R d , dµ k ) .By the Riesz–Thorin interpolation theorem we obtain sup {k T t f k p,dµ k : f ∈ S ( R d ) , k f k p,dµ k ≤ } ≤ , ≤ p ≤ . Let < p < ∞ , /p + 1 /p ′ = 1 . As for p = 1 we get sup {k T t f k p,dµ k : f ∈ S ( R d ) , k f k p,dµ k ≤ } = sup {k T t g k p ′ ,dµ k : g ∈ S ( R d ) , k g k p ′ ,dµ k ≤ } ≤ . (cid:3) For any f ∈ L p ( R + , dν λ ) , ≤ p ≤ ∞ , λ > − / , let us define the Gegenbauer-type translation operator (see, e.g., [34, 35]) R t f ( r ) = c λ Z π f ( p r + t − rt cos ϕ ) sin λ ϕ dϕ, where c λ is defined by (3.7). We have that k R t k p → p ≤ and H λ ( R t f )( r ) = j λ ( tr ) H λ ( f )( r ) , where f ∈ S ( R + ) . Taking into account (2.3) and (3.2), we notethat for λ = λ k the operator R t is a restriction of T t on radial functions, that is,for f ∈ L p ( R + , dν λ k ) , T t f ( | x | ) = R t f ( r ) , r = | x | . We also mention the following useful properties of the generalized translationoperator T t . Lemma 3.3.
Let t ∈ R . (1) If f ∈ L ( R d , dµ k ) , then R R d T t f dµ k = R R d f dµ k . (2) Let r > , f ∈ L p ( R d , dµ k ) , ≤ p < ∞ . If supp f ⊂ B r , then supp T t f ⊂ B r + | t | . If supp f ⊂ R d \ B r , r > | t | , then supp T t f ⊂ R d \ B r −| t | .Proof. Due to the L p -boundedness of T t and density of S ( R d ) in L p ( R d , dµ k ) wecan assume that f ∈ S ( R d ) . D. V. GORBACHEV, V. I. IVANOV, AND S. YU. TIKHONOV (1) Let s > . By integral representation of j λ k ( z ) (see, e.g., [2, Sect. 7.12]) wehave T t ( e − s | ·| )( x ) = R t ( e − s ( · ) )( | x | ) = c λ k Z π e − s ( | x | + t − | x | t cos ϕ ) sin λ k ϕ dϕ = e − s ( | x | + t ) c λ k Z π e s | x | t cos ϕ sin λ k ϕ dϕ = e − s ( | x | + t ) j λ k (2 is | x | t ) , and, in particular, T t ( e − s | ·| )( x ) ≤ e − s ( | x | + t ) e s | x | t = e − s ( | x |− t ) ≤ . Using the self-adjointness of T t , we obtain Z R d T t f ( x ) e − s | x | dµ k ( x ) = Z R d f ( x ) T t ( e − s | ·| )( x ) dµ k ( x ) . Since for any t ∈ R , x ∈ R d , lim s → e − s | x | = lim s → T t ( e − s | ·| )( x ) = 1 , then by Lebesgue’s dominated convergence theorem we derive (1).(2) If supp f ⊂ B r and | x | > r + | t | , then, in light of (3.4) and (3.3), for z ∈ supp σ kx,t and g ∈ G , we have that | z | ≥ | gx | − | z − gx | = | x | − | z − gx | > r and f ( z ) = 0 , which yields T t f ( x ) = 0 .If supp f ⊂ R d \ B r , | x | < r − | t | , then, for z ∈ supp σ kx,t and g ∈ G , we similarlyobtain | z | ≤ | gx | + | z − gx | = | x | + | z − gx | < r , f ( z ) = 0 , and T t f ( x ) = 0 . (cid:3) Let g be a radial function, g ( y ) = g ( | y | ) , where g ( t ) is defined on R + . Notethat by virtue of (2.2)(3.9) k g k p,dµ k = k g k p,dν λk , F k ( g )( y ) = H λ k ( g )( | y | ) . By means of operators T t and τ y define two convolution operators(3.10) ( f ∗ λk g )( x ) = Z ∞ T t f ( x ) g ( t ) dν λ k ( t ) , (3.11) ( f ∗ k g )( x ) = Z R d f ( y ) τ x g ( − y ) dµ k ( y ) . Note that operator (3.10) was defined in [48], while (3.11) was investigated in[48, 53].S. Thangavelu and Yu. Xu [48] proved that if f ∈ L p ( R d , dµ k ) , ≤ p ≤ ∞ , g ∈ L ( R d , dµ k ) , and g is bounded, then(3.12) k ( f ∗ k g ) k p,dµ k ≤ k f k p,dµ k k g k ,dµ k , and if ≤ p ≤ , g ∈ L p rad ( R d , dµ k ) , then, for any y ∈ R d ,(3.13) k τ y g k p,dµ k ≤ k g k p,dµ k . p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 9 Note that additional condition of boundedness g in (3.12) can be omitted. In-deed, by H¨older’s inequality | ( f ∗ k g )( x ) | = (cid:12)(cid:12)(cid:12)Z R d f ( y ) τ x g ( − y ) dµ k ( y ) (cid:12)(cid:12)(cid:12) ≤ (cid:16)Z R d | f ( y ) | p | τ x g ( − y ) | dµ k ( y ) (cid:17) /p (cid:16)Z R d | τ x g ( − y ) | dµ k ( y ) (cid:17) − /p , and by (3.13) for p = 1 we get k ( f ∗ λk g ) k p,dν λk ≤ (cid:16)Z R d Z R d | f ( y ) | p | τ x g ( − y ) | dµ k ( y ) dµ k ( x ) (cid:17) /p k g k − /p ,dµ k = (cid:16)Z R d | f ( y ) | p Z R d | τ − y g ( x ) | dµ k ( x ) dµ k ( y ) (cid:17) /p k g k − /p ,dµ k ≤ k f k p,dµ k k g k ,dµ k . Lemma 3.4. If f ∈ A k , g ∈ L ( R + , dν λ k ) , g ( y ) = g ( | y | ) , then, for any x, y ∈ R d , (3.14) ( f ∗ λk g )( x ) = ( f ∗ k g )( x ) = Z R d τ − y f ( x ) g ( y ) dµ k ( y ) , (3.15) F k ( f ∗ λk g )( y ) = F k ( f ∗ k g )( y ) = F k ( f )( y ) F k ( g )( y ) . Proof.
Using (3.2) and (3.9), we get ( f ∗ λk g )( x ) = Z ∞ T t f ( x ) g ( t ) dν λ k ( t )= Z ∞ Z R d j λ k ( t | y | ) e k ( x, y ) F k ( f )( y ) dµ k ( y ) g ( t ) dν λ k ( t )= Z R d e k ( x, y ) F k ( f )( y ) F k ( g )( y ) dµ k ( y ) , which gives F k ( f ∗ λk g )( y ) = F k ( f )( y ) F k ( g )( y ) . If g ∈ A k , then, by (3.1), ( f ∗ k g )( x ) = Z R d f ( y ) τ x g ( − y ) dµ k ( y )= Z R d f ( y ) Z R d e k ( − y, z ) e k ( x, z ) F k ( g )( z ) dµ k ( z ) dµ k ( y )= Z R d e k ( x, z ) F k ( f )( z ) F k ( g )( z ) dµ k ( z ) . and hence the first equality in (3.14) and the second equality in (3.15) are validfor g ∈ A k .Assuming that g ∈ L ( R + , dν λ ) , ( g n ) ∈ S ( R + ) , g n → g in L ( R d , dµ k ) , andtaking into account (3.8)–(3.11) and (3.13), we arrive at (cid:12)(cid:12) ( f ∗ λk g )( x ) − ( f ∗ k g )( x ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) ( f ∗ λk ( g − ( g n ) ))( x ) (cid:12)(cid:12) + (cid:12)(cid:12) ( f ∗ k ( g − g n ))( x ) (cid:12)(cid:12) ≤ k f k ∞ k g − g n k ,dµ k . Thus, the first equality in (3.14) holds.
Finally, using (3.1), we get Z R d τ − y f ( x ) g ( y ) dµ k ( y ) = Z R d g ( y ) Z R d e k ( − y, z ) e k ( x, z ) F k ( f )( z ) dµ k ( z ) dµ k ( y )= Z R d e k ( x, z ) F k ( f )( z ) F k ( g )( z ) dµ k ( z ) . and the second part in (3.14) is valid. (cid:3) Let y ∈ R d be given. M. R¨osler [42] proved that the operator τ y is positive on C ∞ rad ( R d ) , i.e., τ y ≥ , and moreover, for any x ∈ R d ,(3.16) τ y f ( x ) = Z R d f ( z ) dρ kx,y ( z ) , where ρ kx,y is a radial probability Borel measure such that supp ρ kx,y ⊂ B | x | + | y | . Theorem 3.5. If ≤ p ≤ ∞ , then, for any x ∈ R d and f ∈ S ( R d ) , (3.17) k T t f ( x ) k p,dν λk = (cid:16)Z R + | T t f ( x ) | p dν λ ( t ) (cid:17) /p ≤ k f k p,dµ k . Proof.
Let x ∈ R d be given. Let an operator B x be defined on S ( R d ) as follows(cf. (3.2) and (3.3)): for f ∈ S ( R d ) , B x f ( t ) = T t f ( x ) = Z R d j λ k ( t | y | ) e k ( x, y ) F k ( f )( y ) dµ k ( y ) = Z R d f ( z ) dσ kx,t ( z ) . Let p = 2 . We have T t f ( x ) = Z ∞ j λ k ( tr ) Z S d − e k ( x, ry ′ ) F k ( f )( ry ′ ) dσ k ( y ′ ) dν λ k ( r ) and H λ k ( T t f ( x ))( r ) = Z S d − e k ( x, ry ′ ) F k ( f )( ry ′ ) dσ k ( y ′ ) . This, H¨older’s inequality, and the fact that the operators H λ k and F k are unitaryimply k T t f ( x ) k ,dν λk = kH λ k ( T t f ( x ))( r ) k ,dν λk = Z ∞ (cid:12)(cid:12)(cid:12)Z S d − e k ( x, ry ′ ) F k ( f )( ry ′ ) dσ k ( y ′ ) (cid:12)(cid:12)(cid:12) dν λ k ( r ) ≤ Z ∞ Z S d − |F k ( f )( ry ′ ) | dσ k ( y ′ ) dν λ k ( r )= kF k ( f ) k ,dµ k = k f k ,dµ k , which yields inequality (3.17) for p = 2 . Moreover, B x can be extended to the space L ( R + , dν λ k ) with preservation of norm, and, moreover, this extension coincideswith (3.2).Let p = 1 . By (3.14) and (3.16), we obtain k T t f ( x ) k ,dν λk = sup nZ ∞ T t f ( x ) g ( t ) dν λ k ( t ) : g ∈ S ( R + ) , k g k ∞ ≤ o = sup nZ R d f ( y ) τ x g ( − y ) dµ k ( y ) : g ∈ S rad ( R d ) , k g k ∞ ≤ o ≤ k f k ,dµ k sup (cid:8) k τ x g ( − y ) k ∞ : g ∈ S rad ( R d ) , k g k ∞ ≤ (cid:9) ≤ k f k ,dµ k , p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 11 which is the desired inequality (3.17) for p = 1 . Moreover, B x can be extendedto L ( R + , dν λ k ) with preservation of norm such that the extension coincides with(3.2) on L ( R + , dν λ k ) ∩ L ( R + , dν λ k ) .By the Riesz–Thorin interpolation theorem we obtain (3.17) for < p < .If < p < ∞ , /p + 1 /p ′ = 1 , then by (3.14) and (3.13), k T t f ( x ) k p,dν λk = sup nZ ∞ T t f ( x ) g ( t ) dν λ k ( t ) : g ∈ S ( R + ) , k g k p ′ ,dν λk ≤ o = sup nZ R d f ( y ) τ x g ( − y ) dµ k ( y ) : g ∈ S rad ( R d ) , k g k p ′ ,dµ k ≤ o ≤ k f k p,dµ k sup {k τ x g ( − y ) k p ′ ,dµ k : g ∈ S rad ( R d ) , k g k p ′ ,dµ k ≤ }≤ k f k p,dµ k . Finally, for p = ∞ , (3.17) follows from representation (3.3). (cid:3) We are now in a position to prove the Young inequality for the convolutions(3.10) and (3.11).
Theorem 3.6.
Let ≤ p, q ≤ ∞ , p + q ≥ , and r = p + q − . We have that,for any f ∈ S ( R d ) , g ∈ S ( R + ) and g ∈ S rad ( R d ) , (3.18) k ( f ∗ λk g ) k r,dν λk ≤ k f k p,dµ k k g k q,dν λk , (3.19) k ( f ∗ k g ) k r,dµ k ≤ k f k p,dµ k k g k q,dµ k . Proof.
Since for g ( y ) = g ( | y | ) we have k ( f ∗ λk g ) k r,dν λk = k ( f ∗ k g ) k r,dµ k , k g k q,dν λk = k g k q,dµ k , it is enough to show inequality (3.18). The proof is straightforward using H¨older’sinequality and estimates (3.6) and (3.17). For the sake of completeness, we giveit here. Let µ = p − r and ν = q − r , then µ ≥ , ν ≥ and r + µ + ν = 1 . Invirtue of (3.17), we have (cid:12)(cid:12)(cid:12)Z ∞ T t f ( x ) g ( t ) dν λ k ( t ) (cid:12)(cid:12)(cid:12) ≤ (cid:16)Z ∞ | T t f ( x ) | p | g ( t ) | q dν λ k ( t ) (cid:17) /r × (cid:16)Z ∞ | T t f ( x ) | p dν λ k ( t ) (cid:17) /µ (cid:16)Z ∞ | g ( t ) | q dν λ k ( t ) (cid:17) /ν ≤ (cid:16)Z ∞ | T t f ( x ) | p | g ( t ) | q dν λ k ( t ) (cid:17) /r k f k p/µp,dµ k k g k q/νq,dν λk . Using (3.6), this gives k ( f ∗ λk g ) k r,dν λk ≤ (cid:16)Z R d Z ∞ | T t f ( x ) | p | g ( t ) | q dν λ k ( t ) dµ k ( x ) (cid:17) /r × k f k p/µp,dµ k k g k q/νq,dν λk ≤ k f k p,dµ k k g k q,dν λk . (cid:3) Theorem 3.7.
Let ≤ p ≤ ∞ and g ∈ S rad ( R d ) . We have that, for any y ∈ R d , (3.20) k τ y g k p,dµ k ≤ k g k p,dµ k . Remark . Since S ( R d ) is dense in L p ( R d , dµ k ) , ≤ p < ∞ , the operator τ y canbe defined on L p rad ( R d , dµ k ) so that (3.20) holds. Proof.
In the case ≤ p ≤ this result was proved in [48]. The case p = ∞ follows from (3.16).Let < p < ∞ . Since F k ( g ) is a radial function and τ y g ( − x ) = Z R d e k ( y, z ) e k ( − x, z ) F k ( g )( z ) dµ k ( z )= Z R d e k ( − y, z ) e k ( x, z ) F k ( g )( z ) dµ k ( z ) = τ − y g ( x ) , then using (3.19) for r = ∞ , q = p we obtain k τ − y g k p,dµ k = sup nZ R d τ − y g ( x ) f ( x ) dµ k ( x ) : f ∈ S ( R d ) , k f k p ′ ,dµ k ≤ o ≤ sup {k ( f ∗ k g )( y ) k ∞ ,dµ k : f ∈ S ( R d ) , k f k p ′ ,dµ k ≤ } ≤ k g k p,dµ k . (cid:3) Now we give an analogue of Lemma 3.4 for the case when f ∈ L p . Lemma 3.9.
Let ≤ p ≤ ∞ , f ∈ L p ( R d , dµ k ) ∩ C b ( R d ) ∩ C ∞ ( R d ) , g ∈ S ( R + ) ,and g ( y ) = g ( | y | ) . Then, for any x ∈ R d , (3.21) ( f ∗ λk g )( x ) = ( f ∗ k g )( x ) ∈ L p ( R d , dµ k ) ∩ C b ( R d ) ∩ C ∞ ( R d ) and, in the sense of tempered distributions, (3.22) F k ( f ∗ λk g ) = F k ( f ∗ k g ) = F k ( f ) F k ( g ) . Proof.
First, in light of (3.6) and (3.18), we note that the convolution (3.10)belongs to L p ( R d , dµ k ) . Moreover, (3.3) implies that it is in C b ( R d ) .Taking into account that g ∈ S ( R d ) and ( − ∆ k ) r e k ( · , z ) = | z | r e k ( · , z ) , we have ( − ∆ k ) r ( f ∗ k g )( x ) = Z R d f ( y ) Z R d e k ( x, z ) e k ( − y, z ) | z | r F k ( g )( z ) dµ k ( z ) dµ k ( y ) . Let us show that the integral converges uniformly in x . We have Z R d e k ( x, z ) e k ( − y, z ) | z | r F k ( g )( z ) dµ k ( z ) = τ x G ( − y ) , where G ∈ S rad ( R d ) is such that F k ( G )( z ) = | z | r F k ( g )( z ) . Using H¨older’s in-equality and (3.20), we get (cid:12)(cid:12)(cid:12)Z R d f ( y ) Z R d e k ( x, z ) e k ( − y, z ) | z | r F k ( g )( z ) dµ k ( z ) dµ k ( y ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)Z R d f ( y ) τ x G ( − y ) dµ k ( y ) (cid:12)(cid:12)(cid:12) ≤ k f k p,dµ k k τ x G k p ′ ,dµ k ≤ k f k p,dµ k k G k p ′ ,dµ k . Thus, convolution (3.11) belongs to C ∞ ( R d ) .By Lemma 3.4, the equality in (3.21) holds for any function f ∈ S ( R d ) . If f ∈ L p ( R d , dµ k ) , f n ∈ S ( R d ) and f n → f in L p ( R d , dµ k ) , then Minkowski’s inequalityand (3.6) give(3.23) k (( f − f n ) ∗ λk g ) k p,dµ k ≤ k f − f n k p,dµ k k g k ,dν λk , while H¨older’s inequality and (3.20) imply | (( f − f n ) ∗ k g )( x ) | ≤ k f − f n k p,dµ k k g k p ′ ,dµ k . By (3.23), there is a subsequence { n k } such that ( f n k ∗ λk g )( x ) → ( f ∗ λk g )( x ) a.e., therefore the relation ( f ∗ λk g )( x ) = ( f ∗ k g )( x ) holds almost everywhere.Since both convolutions are continuous, then it holds everywhere. p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 13 To prove the second equation of the lemma, we first remark that Lemma 3.4implies that (3.22) holds pointwise for any f ∈ S ( R d ) . In the general case, since f ∈ L p ( R d , dµ k ) , ( f ∗ λk g ) ∈ L p ( R d , dµ k ) , and F k ( g ) ∈ S ( R d ) , the left and righthand sides of (3.22) are tempered distributions. Recall that the Dunkl transformof tempered distribution is defined by hF k ( f ) , ϕ i = h f, F k ( ϕ ) i , f ∈ S ′ ( R d ) , ϕ ∈ S ( R d ) . Let f n ∈ S ( R d ) and f n → f in L p ( R d , dµ k ) , ϕ ∈ S ( R d ) . Then hF k (( f − f n ) ∗ λk g ) , ϕ i = h (( f − f n ) ∗ λk g ) , F k ( ϕ ) i , hF k ( g ) F k ( f − f n ) , ϕ i = h ( f − f n ) , F k ( F k ( g ) ϕ ) i and |hF k (( f − f n ) ∗ λk g ) , ϕ i| ≤ k f − f n k p,dµ k k g k ,dν λk kF k ( ϕ ) k p ′ ,dµ k , |hF k ( g ) F k ( f − f n ) , ϕ i| ≤ k f − f n k p,dµ k kF k ( F k ( g ) ϕ ) k p ′ ,dµ k . Thus, the proof of (3.22) is now complete. (cid:3) Boundedness of the Riesz potential
Recall that λ k = d/ − P a ∈ R + k ( a ) . For < α < λ k + 2 , the weightedRiesz potential I kα f , is defined on S ( R d ) (see [49]) by I kα f ( x ) = ( d αk ) − Z R d τ − y f ( x ) 1 | y | λ k +2 − α dµ k ( y ) , where d αk = 2 − λ k − α Γ( α/ / Γ( λ k + 1 − α/ . We have, in the sense of tempereddistributions, F k ( I kα f )( y ) = | y | − α F k ( f )( y ) . Using (2.2) and (3.2), we obtain(4.1) I kα f ( x ) = ( d αk ) − Z ∞ T t f ( x ) 1 t λ k +2 − α dν λ k ( t ) . To estimate the L p -norm of this operator, we use the maximal function definedfor f ∈ S ( R d ) as follows ([48]): M k f ( x ) = sup r> | ( f ∗ k χ B r )( x ) | R B r dµ k , where χ B r is the characteristic function of the Euclidean ball B r of radius r cen-tered at 0.Using (2.2), (3.2), and (3.14), we get M k f ( x ) = sup r> | R r T t f ( x ) dν λ k ( t ) | R r dν λ k . It is proved in [48] that the maximal function is bounded on L p ( R d , dµ k ) ,
a } dµ k . k f k ,dµ k a , a > . Theorem 4.1. If < p < q < ∞ , < α < λ k + 2 , p − q = α λ k +2 , then (4.4) k I kα f k q,dµ k . k f k p,dµ k , f ∈ S ( R d ) . The mapping f I kα f is of weak type (1 , q ) , that is, (4.5) Z { x : | I kα f ( x ) | >a } dµ k . (cid:16) k f k ,dµ k a (cid:17) q . Remark . In the case k ≡ , inequality (4.4) was proved by S. Soboleff [44] andG. O. Thorin [50] and the weighted inequality was studied by E. M. Stein andG. Weiss [46]. For the reflection group G = Z d , Theorem 4.1 was proved in [49].The general case was obtained in [21]. We give another simple proof based on the L p -boundedness of T t given in Theorem 3.5 and follow the proof given in [49] for G = Z d . Remark . In Theorem 4.1, dealing with (4.4), we may assume that f ∈ L p ( R d , dµ k ) , < p < ∞ , while proving (4.5), we may assume that f ∈ L ( R d , dµ k ) . Proof.
Let
R > be fixed. We write (4.1) as sum of two terms, I kα f ( x ) = ( d αk ) − Z R T t f ( x ) 1 t λ k +2 − α dν λ k ( t )+ ( d αk ) − Z ∞ R T t f ( x ) 1 t λ k +2 − α dν λ k ( t ) = J + J . (4.6)Integrating J by parts, we obtain d αk J = Z R t − (2 λ k +2 − α ) d (cid:16)Z t T s f ( x ) dν λ k ( s ) (cid:17) = R α · R − (2 λ k +2) Z R T s f ( x ) dν λ k ( s )+ (2 λ k + 2 − α ) Z R t − (2 λ k +2) Z t T s f ( x ) dν λ k ( s ) t α − dt. (4.7)Here we have used that lim ε → ε α · ε − (2 λ k +2) Z ε T s f ( x ) dν λ k ( s ) = 0 , since ε α · ε − (2 λ k +2) (cid:12)(cid:12)(cid:12)Z ε T s f ( x ) dν λ k ( s ) (cid:12)(cid:12)(cid:12) . ε α sup ε> | R ε T t f ( x ) dν λ k ( t ) | R ε dν λ k = ε α M k f ( x ) . In light of (4.7), we have(4.8) | J | . R α M k f ( x ) + Z R M k f ( x ) t α − dt . R α M k f ( x ) . To estimate J , we use H¨older’s inequality, the relation p − q = α λ k +2 and (3.17): | J | ≤ ( d αk ) − (cid:16)Z ∞ R t − (2 λ k +2 − α ) p ′ dν λ k ( t ) (cid:17) /p ′ k T t f ( x ) k p,dν λk . R − (2 λ k +2) q k f k p,dµ k . This, (4.6) and (4.8) yield | I kα f ( x ) | . R α M k f ( x ) + R − (2 λ k +2) q k f k p,dµ k , p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 15 for any R > . Choosing R = (cid:0) M k f ( x ) / k f k p,dµ k (cid:1) − q/ (2 λ k +2) implies the inequality(4.9) | I kα f ( x ) | . ( M k f ( x )) p/q ( k f k p,dµ k ) − p/q for any ≤ p < q . Integrating (4.9) and using (4.2), we have k I kα f k q,dµ k . k M k f k p/qp,dµ k k f k − p/qp,dµ k . k f k p,dµ k , p > . Finally, we use inequality (4.3) for the maximal function and inequality (4.9)with p = 1 to obtain Z { x : | I kα f ( x ) | >a } dµ k ≤ Z { x : ( M k f ( x )) /q ( k f k ,dµk ) − /q & a } dµ k . (cid:16) k f k ,dµ k a (cid:17) q . (cid:3) Entire functions of exponential type andPlancherel–Polya–Boas-type inequalities
Let C d be the complex Euclidean space of d dimensions. Let also z =( z , . . . , z d ) ∈ C d , Im z = (Im z , . . . , Im z d ) , and σ > .In this section we define several classes of entire functions of exponential typeand study their interrelations. Moreover, we prove the Plancherel–Polya–Boas-type estimates and the Paley–Wiener-type theorems. These classes will be usedlater to study the approximation of functions on R d by entire functions of expo-nential type.First, we define two classes of entire functions: B σp,k and e B σp,k . We say that afunction f ∈ B σp,k if f ∈ L p ( R d , dµ k ) is such that its analytic continuation to C d satisfies | f ( z ) | ≤ c ε e ( σ + ε ) | z | , ∀ ε > , ∀ z ∈ C . The smallest σ = σ f in this inequality is called a spherical type of f . In otherwords, the class B σp,k is the collection of all entire functions of spherical type atmost σ .We say that a function f ∈ e B σp,k if f ∈ L p ( R d , dµ k ) is such that its analyticcontinuation to C d satisfies | f ( z ) | ≤ c f e σ | Im z | , ∀ z ∈ C d . Historically, functions from e B σp,k were basic objects in the Dunkl harmonic analysis.It is clear that e B σp,k ⊂ B σp,k . Moreover, if k ≡ , then both classes coincide (see,e.g., [29]). Indeed, if f ∈ B σp, , ≤ p < ∞ , then Nikol’skii’s inequality [31, 3.3.5] k f k ∞ ≤ d σ d/p k f k p,dµ and the inequality [31, 3.2.6] k f ( · + iy ) k ∞ ≤ e σ | y | k f k ∞ , y ∈ R d , imply that, for z = x + iy ∈ C d , | f ( z ) | ≤ d σ d/p k f k p,dµ e σ | Im z | , i.e., f ∈ e B σp, .In fact, the classes B σp,k and e B σp,k coincide in the weighted case ( k = 0 ) as well.To see that it is enough to show that functions from B σp,k are bounded on R d . Theorem 5.1. If < p < ∞ , then B σp,k = e B σp,k . We will actually prove the more general statement. Let m ∈ Z + , α , . . . , α m ∈ R d \ { } , k ≥ , k , . . . , k m > , and(5.1) v ( x ) = | x | k m Y j =1 |h α j , x i| k j be the power weight. The Dunkl weight is a particular case of such weighted func-tions. The weighted function (5.1) arises in the study of the generalized Fouriertransform (see, e.g., [3]).Let L p,v ( R d ) , < p < ∞ , be the space of complex-valued Lebesgue measurablefunctions f for which k f k p,v = (cid:16)Z R d | f ( x ) | p v ( x ) dx (cid:17) /p < ∞ . Let σ = ( σ , . . . , σ d ) , σ , . . . , σ d > . Again, let us define three anisotropic classes of entire functions: B σ , B σ p,v , and e B σ p,v .We say that a function f defined on R d belongs to B σ if its analytic continuationto C d satisfies | f ( z ) | ≤ c ε e ( σ + ε ) | z | + ··· +( σ d + ε ) | z d | , ∀ ε > , ∀ z ∈ C d . We say that a function f ∈ B σ p,v if f ∈ L p ( R d , dµ k ) is such that its analyticcontinuation to C d belongs to B σ .We say that a function f ∈ e B σ p,v if f ∈ L p ( R d , dµ k ) is such that its analyticcontinuation to C d satisfies | f ( z ) | ≤ c f e σ | Im z | + ··· + σ d | Im z d | , ∀ z ∈ C d . We will use the notation L p ( R d ) , k · k p , B σ p and e B σ p in the case of the unit weight,i.e., v ≡ . Theorem 5.2. If < p < ∞ , then (1) B σ p,v ⊂ B σ p , (2) B σ p,v = e B σ p,v , (3) B σp,v = e B σp,v .Remark . (i) Part (3) of Theorem 5.2 implies Theorem 5.1.(ii) Note that in some particular cases ( k = 0 and p ≥ ) a similar result wasdiscussed in [23].Parts (2) and (3) of Theorem 5.2 follows from (1). Indeed, the embedding in(1) implies that B σ p,v ⊂ B σ p ⊂ B σ ∞ . Hence, a function f ∈ B σ p,v is bounded on R d and then f ∈ e B σ p,v , which gives (2). Further, there holds B σp,v ⊂ B σ p,v , where σ = ( σ, . . . , σ ) ∈ R d + since | z | ≤ | z | + · · · + | z d | . Hence, similar to the above, wehave B σp,v ⊂ B σ ∞ and (3) follows. Thus, to prove Theorem 5.2, it is sufficient toverify part (1).The main difficulty to prove Theorem 5.2 is that the weight v ( x ) vanishes. In or-der to overcome this problem we will first prove two-sided estimates of the L p normof an entire functions in terms of the weighted l p norm, (cid:0)P n v ( λ ( n ) ) | f ( λ ( n ) ) | p (cid:1) /p , < p < ∞ , where v does not vanish at { λ ( n ) } ⊂ R d . p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 17 Such estimates are of their own interest. They generalize the Plancherel–Polyainequality ([33], [6, Chapt. 6, 6.7.15]) X k ∈ Z | f ( λ k ) | p ≤ c ( δ, σ, p ) Z ∞−∞ | f ( x ) | p dx, < p < ∞ , where λ k is increasing sequence such that λ k +1 − λ k ≥ δ > , and f is an entirefunction of exponential type at most σ , and the Boas inequality [5], [6, Chapt. 10,10.6.8],(5.2) Z ∞−∞ | f ( x ) | p dx ≤ C ( δ, L, σ, p ) X k ∈ Z | f ( λ k ) | p , < p < ∞ , where, additionally, (cid:12)(cid:12) λ k − πσ k (cid:12)(cid:12) ≤ L and the type of f is < σ .We write σ ′ = ( σ ′ , . . . , σ ′ d ) < σ = ( σ , . . . , σ d ) if σ ′ < σ , . . . , σ ′ d < σ d . Let n = ( n , . . . , n d ) ∈ Z d and λ ( n ) : Z d → R d . In what follows we consider thesequences of the following type:(5.3) λ ( n ) = ( λ ( n ) , λ ( n , n ) , . . . , λ d ( n , . . . , n d )) , where λ ( n ) i = λ i ( n , . . . , n i ) are sequences increasing with respect to n i , i = 1 , . . . , d for fixed n , . . . , n i − . Definition . We say that the sequence λ ( n ) satisfies the separation condition Ω sep [ δ ] , δ > , if, for any n ∈ Z d , λ i ( n , . . . , n i − , n i + 1) − λ i ( n , . . . , n i − , n i ) ≥ δ, i = 1 , . . . , d. Note that if the sequence λ ( n ) satisfies the separation condition Ω sep [ δ ] then italso satisfies the condition inf n = m | λ ( n ) − λ ( m ) | > . Definition . We say that the sequence λ ( n ) satisfies the close-lattice condition Ω lat [ a , L ] , a = ( a , . . . , a d ) > , L > , if, for any n ∈ Z d , (cid:12)(cid:12)(cid:12) λ i ( n , . . . , n i ) − πn i a i (cid:12)(cid:12)(cid:12) ≤ L, i = 1 , . . . , d.
We start with the Plancherel–Polya-type inequality.
Theorem 5.6.
Assume that λ ( n ) satisfies the condition inf n = m | λ ( n ) − λ ( m ) | > .Then for f ∈ B σ p , < p < ∞ , we have X n ∈ Z d | f ( λ ( n ) ) | p . Z R d | f ( x ) | p dx. Proof.
For the simplicity we prove this result for d = 2 . The proof in the generalcase is similar.The function | f ( z ) | p is plurisubharmonic, and therefore for any x = ( x , x ) ∈ R one has [38] | f ( x , x ) | p ≤ π ) Z π Z π | f ( x + ρ e iθ , x + ρ e iθ | p dθ dθ , where ρ , ρ > . Following [31, 3.2.5], for δ > and ξ + iη = ( ξ + iη , ξ + iη ) we obtain that(5.4) | f ( x , x ) | p ≤ πδ ) Z δ − δ Z δ − δ Z x + δx − δ Z x + δx − δ | f ( ξ + iη ) | p dξ dξ dη dη . The separation condition implies that for some δ > the boxes [ λ ( n )1 − δ, λ ( n )1 + δ ] × [ λ ( n )2 − δ, λ ( n )2 + δ ] do not overlap for any n .Since f ( x + iy ) = X k ∈ Z f ( k ) ( x ) k ! ( iy ) k , where f ( k ) is a partial derivative f of order k = ( k , k ) , k ! = k ! k ! , and ( iy ) k =( iy ) k ( iy ) k , then, applying Bernstein’s inequality (see [31, 3.2.2 and 3.3.5] and[37]), we derive that k f ( · + iy ) k p . e σ | y | + σ | y | k f k p . Using this and (5.4) we derive that X n ∈ Z | f ( λ ( n ) ) | p ≤ πδ ) Z δ − δ Z δ − δ Z ∞−∞ Z ∞−∞ | f ( ξ + iη ) | p dξ dξ dη dη . Z δ − δ Z δ − δ e p ( σ | η | + σ | η d | ) dη dη Z ∞−∞ Z ∞−∞ | f ( ξ ) | p dξ dξ . Z R | f ( x ) | p dx. (cid:3) Theorem 5.7.
Let the sequence λ ( n ) of form (5.3) satisfy the conditions Ω sep [ δ ] and Ω lat [ σ , L ] . Assume that f ∈ B σ ′ , σ ′ < σ , is such that P n ∈ Z d | f ( λ ( n ) ) | p < ∞ , < p < ∞ . Then f ∈ L p ( R d ) and Z R d | f ( x ) | p dx . X n ∈ Z d | f ( λ ( n ) ) | p . Remark . For p ≥ , a similar two-sided Plancherel–Polya–Boas-type inequalitywas obtained from [32]. Proof.
For the simplicity we consider the case d = 2 . Integrating | f ( x , x ) | p at x and applying inequality (5.2), we get, for any x , Z ∞−∞ | f ( x , x ) | p dx . X n ∈ Z | f ( β ( n ) , x ) | p . Since by (5.2), for any n , Z ∞−∞ | f ( β ( n ) , x ) | p dx . X n ∈ Z | f ( β ( n ) , β ( n , n )) | p , then Z ∞−∞ Z ∞−∞ | f ( x , x ) | p dx dx . X n ∈ Z Z ∞−∞ | f ( β ( n ) , x ) | p dx . X n ∈ Z X n ∈ Z | f ( β ( n ) , β ( n , n )) | p < ∞ . (cid:3) Using Theorems 5.6 and 5.7 we arrive at the following statement. p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 19 Theorem 5.9.
Let the sequence { λ ( n ) } of form (5.3) satisfy the conditions Ω sep [ δ ] and Ω lat [ σ , L ] . If f ∈ B σ ′ , σ ′ < σ , then, for < p < ∞ , X n ∈ Z d | f ( λ ( n ) ) | p . Z R d | f ( x ) | p dx . X n ∈ Z d | f ( λ ( n ) ) | p . We will need the weighted version of the Plancherel–Polya–Boas equivalence.We start with three auxiliary lemmas.
Lemma 5.10. [18] If γ ≥ − / , then there exists an even entire function ω γ ( z ) , z ∈ C , of exponential type such that, uniformly in x ∈ R + , ω γ ( x ) ≍ ( x k +2 , ≤ x ≤ ,x γ +1 , x ≥ , where k = [ γ + 1 / and [ a ] is the integral part of a . In particular, we can take ω ( z ) = z k +2 j k − γ ( z + i ) j k − γ ( z − i ) . Lemma 5.11.
Let m ∈ N , j = 1 , . . . , m , b j = ( b j , . . . , b jd ) ∈ R d \ { } , and either | b ji | ≥ , or b ji = 0 , i = 1 , . . . , d . Then there exists a sequence { ρ ( n ) } ⊂ Z d \ { } ofthe form (5.3) such that, for any j = 1 , . . . , m and i = 1 , . . . , d , (5.5) | ρ i ( n , . . . , n i ) − n i | ≤ m, (5.6) |h b j , ρ ( n ) i| ≥ / . Proof.
To construct a desired sequence ρ ( n ) = ( ρ ( n ) , ρ ( n , n ) , . . . , ρ d ( n , . . . , n d )) ∈ Z d , we will use the following simple remark. If we throw out m points from Z , thenthe rest can be numbered such that the obtained sequence will be increasing and(5.5) holds.Let J = { j : b j = 0 , b j = · · · = b jd = 0 } . If J = ∅ , then we set ρ ( n ) = n .If J = ∅ , then ρ ( n ) is increasing sequence formed from Z \ { } . In both cases(5.5) is valid and, moreover, for j ∈ J and any ρ ( n , n ) , . . . , ρ d ( n , . . . , n d ) , onehas (5.6) since |h b j , ρ ( n ) i| = | b j ρ ( n ) | ≥ .Let J = { j : b j = 0 , b j = · · · = b jd = 0 } , n ∈ Z . If J = ∅ , then we set ρ ( n , n ) = n . Let J = ∅ . If j ∈ J and b j ρ ( n ) + b j t j = 0 , then t j = l j + ε j , l j ∈ Z , | ε j | ≤ / . Here l j is the nearest integer to t j . Note that if ρ = l j , then | b j ρ ( n ) + b j ρ | = | b j ( ρ − l j − ε j ) | ≥ / .Let ρ ( n , n ) be an increasing sequence at n formed from Z \{ l j : j ∈ J } . Forthis sequence (5.5) holds and, for j ∈ J and any ρ ( n , n , n ) , . . . , ρ d ( n , . . . , n d ) ,one has |h b j , ρ ( n ) i| = | b j ρ ( n ) + b j ρ ( n , n ) | ≥ / , that is, (5.6) holds as well.Assume that we have constructed the sets J , . . . , J d − , and the sequence ( ρ ( n ) , ρ ( n , n ) , . . . , ρ d − ( n , . . . , n d − )) ∈ Z d − .Let J d = { j : b jd = 0 } , ( n , . . . , n d − ) ∈ Z d − . If J d = ∅ , then we set ρ d ( n , . . . , n d − , n d ) = n d . Assume now that J d = ∅ . If j ∈ J d and b j ρ ( n ) + · · · + b jd − ρ d − ( n , . . . , n d − ) + b jd t j = 0 , then t j = l j + ε j , | ε j | ≤ / . Note that if ρ d = l j , then | b j ρ ( n ) + · · · + b jd − ρ d − ( n , . . . , n d − ) + b jd ρ d | = | b jd ( ρ d − l j − ε j ) | ≥ / . Let ρ d ( n , . . . , n d ) be an increasing sequence in n d formed from Z \ { l j : j ∈ J d } , ρ ( n ) = ( ρ ( n ) , ρ ( n , n ) , . . . , ρ d ( n , . . . , n d )) . For the sequence ρ d ( n , . . . , n d ) inequality (5.5) holds and, for j ∈ J d , one has |h b j , ρ ( n ) i| ≥ / .Thus, we construct the desired sequence since, for any j ∈ { , . . . , m } and some i ∈ { , . . . , d } , there holds b j ∈ J i . (cid:3) An important ingredient of the proof of Theorem 5.2 is the following corollaryof Lemma 5.11.
Lemma 5.12. If a > , α , . . . , α m ∈ R d \ { } , then there exists a sequence λ ( n ) of the form (5.3) such that for some δ, L > the conditions Ω sep [ δ ] , Ω lat [ a , L ] , and ξ j ( λ ( n ) ) ≥ δ , j = 0 , , . . . , m , n ∈ Z d , hold, where (5.7) ξ ( x ) = | x | , ξ j ( x ) = |h α j , x i| , j = 1 , . . . , m. Indeed, for m ≥ it is enough to define λ ( n ) = ( λ ( n ) , λ ( n , n ) , . . . , λ d ( n , . . . , n d )):= (cid:16) πρ ( n ) a , πρ ( n , n ) a , . . . , πρ d ( n , . . . , n d ) a d (cid:17) , (5.8)where ρ ( n ) is the sequence defined in Lemma 5.11. For m = 0 in (5.8), we can take { ρ ( n ) } = Z d \ { } .We are now in a position to state the Plancherel–Polya–Boas inequalities withweights. Theorem 5.13.
Let f ∈ B σ and λ ( n ) be the sequence satisfying all conditions ofLemma 5.12 with some a > σ , then, for < p < ∞ , X n ∈ Z d v ( λ ( n ) ) | f ( λ ( n ) ) | p . Z R d | f ( x ) | p v ( x ) dx . X n ∈ Z d v ( λ ( n ) ) | f ( λ ( n ) ) | p . Proof.
Recall that v ( x ) = Q mj =0 v j ( x ) , where v j ( x ) = ξ k j j ( x ) , j = 0 , , . . . , m (see(5.1) and (5.7)).By Lemma 5.10, we construct entire function of exponential type w ( z ) = m Y j =0 w j ( z ) , where w ( z ) = ω γ ( | z | ) , w j ( z ) = ω γ j ( h α j , z i ) , j = 1 , . . . , m , and γ j = k j p − , j = 0 , , . . . , m. For j = 0 , , . . . , m , we have w j ∈ B µ j , where µ = (1 , . . . , ∈ R d , µ j = ( | α j | , . . . , | α jd | ) , j = 1 , . . . , m, and w ∈ B µ , µ = P mj =0 µ j . Moreover, for any j = 0 , , . . . , m ,(5.9) w pj ( x ) . v j ( x ) , x ∈ R d ,w pj ( x ) & v j ( x ) & , for ξ j ( x ) ≥ δ > . p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 21 Let f ∈ B σ p,v , < p < ∞ , σ < a , and λ ( n ) be the sequence satisfying allconditions of Lemma 5.12, then, for some s > such that σ + 2 s µ < a , we havethat f ( x ) w ( sx ) ∈ B σ +2 s µ .Using Theorem 5.6, and properties (5.9), we derive X n ∈ Z d v ( λ ( n ) ) | f ( λ ( n ) ) | p . X n ∈ Z d | f ( λ ( n ) ) w ( λ ( n ) ) | p . Z R d | f ( x ) w ( x ) | p dx . Z R d | f ( x ) | p v ( x ) dx. Let δ > , J ⊂ J m := { , , . . . , m } or J = ∅ , E δ ( J ) = { x ∈ R d : ξ j ( x ) ≥ δ, j ∈ J and ξ j ( x ) ≤ δ, j ∈ J m \ J } . Since f ( x ) Q j ∈ J w j ( sx ) ∈ B σ +2 s µ , then using Theorems 5.6, 5.7 and properties(5.9) for δ from Lemma 5.12, we obtain Z R d | f ( x ) | p v ( x ) dx = X J Z E δ ( J ) | f ( x ) | p v ( x ) dx . X J Z E δ ( J ) | f ( x ) | p Y j ∈ J v j ( sx ) dx . X J Z E δ ( J ) (cid:12)(cid:12)(cid:12) f ( x ) Y j ∈ J w j ( sx ) (cid:12)(cid:12)(cid:12) p dx . X J Z R d (cid:12)(cid:12)(cid:12) f ( x ) Y j ∈ J w j ( sx ) (cid:12)(cid:12)(cid:12) p dx . X n | f ( λ ( n ) ) | p X J Y j ∈ J w pj ( sλ ( n ) ) . X n | f ( λ ( n ) ) w ( sλ ( n ) ) | p . X n | f ( λ ( n ) ) | p v ( sλ ( n ) ) . X n | f ( λ ( n ) ) | p v ( λ ( n ) ) , where we have assumed that Q j ∈ ∅ = 1 . (cid:3) Proof of Theorem 5.2.
Recall that it is enough to show that B σ p,v ⊂ B σ p and thelatter follows from B σ p,v ⊂ L p ( R d ) .Let f ∈ B σ p,v , < p < ∞ , a > σ , and λ ( n ) be the sequence satisfying all condi-tions of Lemma 5.12. Using Theorem 5.6, and properties (5.9) as in Theorem 5.13we have Z R d | f ( x ) | p dx . X n ∈ Z d | f ( λ ( n ) ) | p . X n ∈ Z d | w ( λ ( n ) ) f ( λ ( n ) ) | p . Z R d | f ( x ) w ( x ) | p dx . Z R d | f ( x ) | p v ( x ) dx. (cid:3) By the Paley–Wiener theorem for tempered distributions (see [25, 53]) andTheorem 5.1, we arrive at the following result.
Theorem 5.14.
A function f ∈ B σp,k , ≤ p < ∞ , if and only if f ∈ L p ( R d , dµ k ) ∩ C b ( R d ) and supp F k ( f ) ⊂ B σ . The Dunkl transform F k ( f ) in Theorem 5.14 is understood as a function for ≤ p ≤ and as a tempered distribution for p > .We conclude this section by presenting the concept of the best approximation.Let E σ ( f ) p,dµ k = inf {k f − g k p,dµ k : g ∈ B σp,k } be the best approximation of a function f ∈ L p ( R d , dµ k ) by entire functions ofspherical exponential type σ . We show that the best approximation is achieved. Theorem 5.15.
For any f ∈ L p ( R d , dµ k ) , ≤ p ≤ ∞ , there exist a function g ∗ ∈ B σp,k such that E σ ( f ) p,dµ k = k f − g ∗ k p,dµ k .Proof. The proof is standard. Let g n be a sequence from B σp,k such that k f − g n k p,dµ k → E σ ( f ) p,dµ k . Since it is bounded in L p ( R d , dµ k ) , then it is also boundedin C b ( R d ) . A compactness theorem for entire functions [31, 3.3.6] implies thatthere exist a subsequence g n k and an entire function g ∗ of exponential type atmost σ such that lim k →∞ g n k ( x ) = g ∗ ( x ) , x ∈ R d , and, moreover, convergence is uniform on compact sets. Therefore, for any R > , k g ∗ χ B R k p,dµ k = lim k →∞ k g n k χ B R k p,dµ k ≤ M. Letting R → ∞ , we have that g ∗ ∈ B σp,k . In light of k ( f − g ∗ ) χ B R k p,dµ k = lim k →∞ k ( f − g n k ) χ B R k p,dµ k ≤ lim k →∞ k f − g n k k p,dµ k = E σ ( f ) p,dµ k , we have k f − g ∗ k p,dµ k ≤ E σ ( f ) p,dµ k . (cid:3) Jackson’s inequality and equivalence of modulus of smoothnessand K -functional Smoothness characteristics and K -functional. Let S ′ ( R d ) be the spaceof tempered distributions, r ∈ N . We can multiply tempered distributions onfunctions from C ∞ Π ( R d ) . Observe that | x | r , e k ( x, · ) , j λ k ( | x | ) ∈ C ∞ Π ( R d ) .Further, using multipliers from C ∞ Π ( R d ) , Laplacian, and Dunkl transform, wedefine several functionals on the Schwartz space. If a sequence { ϕ l } ⊂ S ( R d ) converges to zero in topology of S ( R d ) , then the sequences { ( − ∆ k ) r ϕ l } , {F k ( ϕ l ) } , { gϕ l } , g ∈ C ∞ Π ( R d ) , also converge to zero in topology of S ( R d ) . Hence all func-tionals will be continue.First we define the Dunkl transform for tempered distribution hF k ( f ) , ϕ i = h f, F k ( ϕ ) i , f ∈ S ′ ( R d ) , ϕ ∈ S ( R d ) . If ˇ ϕ ( y ) = ϕ ( − y ) , the inverse Dunkl transform will be hF − k ( f ) , ϕ i = h f, F − k ( ϕ ) i = h f, F k ( ˇ ϕ ) i , f ∈ S ′ ( R d ) , ϕ ∈ S ( R d ) . Let f, g ∈ S ′ ( R d ) . We have F − k ( F k ( f )) = F k ( F − k ( f )) , and f = g iff F k ( f ) = F k ( g ) .We define the r -th power of the Dunkl Laplacian ( − ∆ k ) r f as follows h ( − ∆ k ) r f, ϕ i = h f, ( − ∆ k ) r ϕ i = h f, F − k | · | r F k ( ϕ )) i , f ∈ S ′ ( R d ) , ϕ ∈ S ( R d ) . Let W rp,k be the Sobolev space, that is, W rp,k = { f ∈ L p ( R d , dµ k ) : ( − ∆ k ) r f ∈ L p ( R d , dµ k ) } equipped with the Banach norm k f k W rp,k = k f k p,dµ k + k ( − ∆ k ) r f k p,dµ k . Note that ( − ∆ k ) r f ∈ S ( R d ) whenever f ∈ S ( R d ) . p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 23 For f ∈ S ′ ( R d ) the generalized translation operators τ y f, T t f ∈ S ′ ( R d ) aredefined as follows h τ y f, ϕ i = h f, τ − y ϕ i = h f, F − k ( e k ( − y, · ) F k ( ϕ )) i , ϕ ∈ S ( R d ) , y ∈ R d , h T t f, ϕ i = h f, T t ϕ i = h f, F − k ( j λ k ( t | · | ) F k ( ϕ )) i , ϕ ∈ S ( R d ) , t ∈ R + . For the Dunkl transform of the considered operators and their compositions wehave the following easily verifiable equalities(6.1) F k (( − ∆ k ) r f ) = | · | r F k ( f ) , F k ( τ y f ) = e k ( y, · ) F k ( f ) , F k (( − ∆ k ) r τ y f ) = | · | r e k ( y, · ) F k ( f ) , F k ( T t f ) = j λ k ( t | · | ) F k ( f ) , F k (( − ∆ k ) r T t f ) = | · | r j λ k ( t | · | ) F k ( f ) , F k ( T t ( τ y f )) = j λ k ( t | · | ) e k ( y, · ) F k ( f ) . This implies the commutativity of these compositions.Let ϕ ∈ S ( R d ) , ˇ ϕ ( y ) = ϕ ( − y ) . We call f ∈ S ′ ( R d ) even if h f, ˇ ϕ i = h f, ϕ i . Notethat f ∈ S ′ ( R d ) is even iff F ( f ) is even.Let N k be a set of even f ∈ S ′ ( R d ) for which F k ( f ) ∈ C ∞ Π ( R d ) . For f ∈ N k and ϕ ∈ S ( R d ) we set ( f ∗ k ϕ )( x ) = h τ x f, ˇ ϕ i = h f, τ − x ˇ ϕ i . If g ∈ N k , ϕ ∈ S ( R d ) , then ( g ∗ k ϕ ) ∈ S ( R d ) and(6.2) F k ( g ∗ k ϕ )( y ) = F k ( g )( y ) F k ( ϕ )( y ) . Indeed, we have τ − x ˇ ϕ ( y ) = F − k ( e k ( − x, · ) F k ( ˇ ϕ ))( y )= Z R d e k ( − x, z ) e k ( y, z ) F k ( ˇ ϕ )( z ) dµ k ( z )= Z R d e k ( − x, z ) e k ( y, z ) F k ( ϕ )( − z ) dµ k ( z )= Z R d e k ( x, z ) e k ( − y, z ) F k ( ϕ )( z ) dµ k ( z )= F k ( e k ( x, · ) F k ( ϕ ))( y ) = τ x ϕ ( − y ) ∈ S ( R d ) . Hence, by definition we get ( g ∗ k ϕ )( x ) = h g, τ − x ˇ ϕ i = h g, F k ( e k ( x, · ) F k ( ϕ )) i = hF k ( g ) , e k ( x, · ) F k ( ϕ ) i = Z R d e k ( x, z ) F k ( g )( z ) F k ( ϕ )( z ) dµ k ( z )= F − k ( F k ( g ) F k ( ϕ ))( x ) ∈ S ( R d ) and the equality (6.2).Now we can define a convolution ( f ∗ k g ) ∈ S ′ ( R d ) for f ∈ S ′ ( R d ) and g ∈ N k as follows(6.3) h ( f ∗ k g ) , ϕ i = h f, ( g ∗ k ϕ ) i , ϕ ∈ S ( R d ) . Applying (6.2) and F k ( F k ( ϕ )) = ˇ ϕ for ϕ ∈ S ( R d ) , we obtain hF k ( f ∗ k g ) , ϕ i = h ( f ∗ k g ) , F k ( ϕ ) i = h f, ( g ∗ k F k ( ϕ )) i = h f, F − k ( F k ( g ) F k ( F k ( ϕ ))) i = h f, F − k ( F k ( g ) ˇ ϕ ) i = h f, F k ( F k ( g ) ϕ ) i = hF k ( f ) , F k ( g ) ϕ i = hF k ( g ) F k ( f ) , ϕ i , hence(6.4) F k ( f ∗ k g ) = F k ( g ) F k ( f ) . The distribution | · | r ∈ S ′ ( R d ) is even, G r = F − k ( | · | r ) ∈ N k and ( − ∆) r f =( f ∗ k G r ) for f ∈ S ′ ( R d ) .If g , g ∈ N k , then ( g ∗ k g ) ∈ N k and ( g ∗ k g ) = ( g ∗ k g ) .If f ∈ S ′ ( R d ) and g , g ∈ N k then ( f ∗ k ( g ∗ k g )) = (( f ∗ k g ) ∗ k g ) . We have ( − ∆ k ) r ( f ∗ k g ) = (( − ∆ k ) r f ∗ k g ) . Indeed, by (6.1), (6.4), F k (( − ∆ k ) r ( f ∗ k g )) = | · | r F k ( f ∗ k g ) = | · | r F k ( f ) F k ( g )= F k (( − ∆ k ) r f ) F k ( g ) = F k (( − ∆ k ) r f ∗ k g ) . Lemma 6.1. If f ∈ L p ( R d , dµ k ) , g ∈ L ( R d , dµ k ) , F k ( g ) ∈ N k , then both con-volutions (3.11) and (6.3) of these functions coincide.Proof. Set ( f ∗ k g )( x ) = Z R d f ( y ) τ x g ( − y ) dµ k ( y ) . By (3.12) ( f ∗ k g ) ∈ L p ( R d , dµ k ) and ( f ∗ k g ) ∈ S ′ ( R d ) . It is sufficiently to provethe equality F k ( f ∗ k g ) = F k ( g ) F k ( f ) in S ′ ( R d ) . For any ϕ ∈ S ( R d ) we have hF k ( f ∗ k g ) , ϕ i = h ( f ∗ k g ) , F k ( ϕ ) i = Z R d Z R d f ( y ) τ x g ( − y ) dµ k ( y ) F k ( ϕ )( x ) dµ k ( x )= Z R d f ( y ) Z R d τ − y g ( x ) F k ( ϕ )( x ) dµ k ( x ) dµ k ( y ) . Since Z R d τ − y g ( x ) F k ( ϕ )( x ) dµ k ( x )= Z R d Z R d e k ( − y, z ) e k ( x, z ) F k ( g )( z ) dµ k ( z ) F k ( ϕ )( x ) dµ k ( x )= Z R d e k ( − y, z ) F k ( g )( z ) ϕ ( z ) dµ k ( z ) = F k ( F k ( g ) ϕ )( y ) , then hF k ( f ∗ k g ) , ϕ i = Z R d f ( y ) F k ( F k ( g ) ϕ ) dµ k = h f, F k ( F k ( g ) ϕ ) i = hF k ( f ) , F k ( g ) ϕ i = hF k ( g ) F k ( f ) , ϕ i . Lemma 6.1 is proved. (cid:3)
Define the K -functional for the couple ( L p ( R d , dµ k ) , W rp,k ) as follows K r ( t, f ) p,dµ k = inf {k f − g k p,dµ k + t r k ( − ∆ k ) r g k p,dµ k : g ∈ W rp,k } . Note that for any f , f ∈ L p ( R d , dµ k ) and g ∈ W rp,k , we have k f − g k p,dµ k + t r k ( − ∆ k ) r g k p,dµ k ≤ k f − g k p,dµ k + t r k ( − ∆ k ) r g k p,dµ k + k f − f k p,dµ k p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 25 and hence,(6.5) | K r ( t, f ) p,dµ k − K r ( t, f ) p,dµ k | ≤ k f − f k p,dµ k . If f ∈ W rp,k , then K r ( t, f ) p,dµ k ≤ t r k ( − ∆ k ) r f k p,dµ k and lim t → K r ( t, f ) p,dµ k = 0 .This and (6.5) imply that, for any f ∈ L p ( R d , dµ k ) ,(6.6) lim t → K r ( t, f ) p,dµ k = 0 . Another important property of the K -functional is(6.7) K r ( λt, f ) p,dµ k ≤ max { , λ r } K r ( t, f ) p,dµ k . Let I be an identical operator and m ∈ N . Consider the following three differ-ences:(6.8) ∆ mt f ( x ) = ( I − T t ) m f ( x ) = m X s =0 ( − s (cid:18) ms (cid:19) ( T t ) s f ( x ) , (6.9) ∗ ∆ mt f ( x ) = m X s =0 ( − s (cid:18) ms (cid:19) T st f ( x ) , (6.10) ∗∗ ∆ mt f ( x ) = (cid:18) mm (cid:19) − m X s = − m ( − s (cid:18) mm − s (cid:19) T st f ( x ) . Differences (6.8) and (6.9) coincide with the classical difference for the translationoperator T t f ( x ) = f ( x + t ) and correspond to the usual definition of the modulusof smoothness of order m . Difference (6.10) can be seen as follows. Define µ s =( − s (cid:0) ms (cid:1) , s ∈ Z . Then the convolution µ ∗ µ is given by ν s := ( µ ∗ µ ) s = X l ∈ Z µ l µ s + l = ( − s (cid:18) mm − s (cid:19) . Note that ν s = 0 if | s | ≤ m . Moreover, if k ≡ , then ν m X s = − m ν s T st f ( x ) = f ( x ) + 2 ν m X s =1 ν s S st f ( x ) = f ( x ) − V m,t f ( x ) , where the operator S t was given in (3.5) and the averages V m,t f ( x ) = − ν m X s =1 ν s S st f ( x ) were defined by F. Dai and Z. Ditzian in [9]. Definition . The moduli of smoothness of a function f ∈ L p ( R d , dµ k ) are definedby(6.11) ω m ( δ, f ) p,dµ k = sup
Since for f ∈ S ′ ( R d ) by (6.1), (6.15), F k ( ∆ m + rt f ) = (1 − j λ k ( t )) m + r F k ( f )= (1 − j λ k ( t )) r (1 − j λ k ( t )) m F k ( f )= r X s =0 ( − s (cid:18) rs (cid:19) ( j λ k ( t )) s F k ( ∆ mt f ) , then ∆ m + rt f = r X s =0 ( − s (cid:18) rs (cid:19) ( T t ) s ( ∆ mt f ) . Using for f ∈ L p ( R d , dµ k ) Theorem 3.5 and (6.14), we get k ∆ m + rt f k p,dµ k ≤ r X s =0 (cid:18) rs (cid:19) k ∆ mt f k p,dµ k = 2 r k ∆ mt f k p,dµ k . (cid:3) Main results.
First we state the Jackson-type inequality.
Theorem 6.6.
Let σ > , ≤ p ≤ ∞ , r ∈ Z + , m ∈ N . We have, for any f ∈ W rp,k , (6.19) E σ ( f ) p,dµ k . σ r Ω m (cid:16) σ , ( − ∆ k ) r f (cid:17) p,dµ k , where Ω m is any of the three moduli of smoothness (6.11) – (6.13) .Remark . (i) For radial functions inequality (6.19) is the Jackson inequality in L p ( R + , dν λ k ) . In this case it was obtained in [34, 35] for moduli (6.11) and (6.12).For k ≡ and the modulus of smoothness (6.13), inequality (6.19) was obtainedby F. Dai and Z. Ditzian [9], see also the paper [10].(ii) From the proof of Theorem 6.6 we will see that inequality (6.19) for moduli(6.11) and (6.13) can be equivalently written as E σ ( f ) p,dµ k . σ r k ∆ m /σ (( − ∆ k ) r f ) k p,dµ k ,E σ ( f ) p,dµ k . σ r k ∗∗ ∆ m /σ (( − ∆ k ) r f ) k p,dµ k . The next theorem provides an equivalence between moduli of smoothness andthe K -functional. Theorem 6.8. If δ > , ≤ p ≤ ∞ , r ∈ N , then for any f ∈ L p ( R d , dµ k ) (6.20) K r ( δ, f ) p,dµ k ≍ ω r ( δ, f ) p,dµ k ≍ ∗∗ ω r ( δ, f ) p,dµ k ≍ ∗ ω r − ( δ, f ) p,dµ k ≍ ∗ ω r ( δ, f ) p,dµ k . Remark . If k ≡ , the equivalence between the classical modulus of smooth-ness and the K -functional is well known [26, 8], while the equivalence betweenmodulus (6.13) and the K -functional was shown in [9]. For radial functions apartial result of (6.20), more precisely, an equivalence of the K -functional andmoduli of smoothness (6.11) and (6.12) was proved in [34, 35]. Remark . One can continue equivalence (6.20) as follows (see also Remark6.17) . . . ≍ k ∆ rδ f k p,dµ k ≍ k ∗∗ ∆ rδ f k p,dµ k . We give the proof for the difference (6.10) and the modulus of smoothness (6.13).We partially follow the proofs in [27, 34, 35] which are different from those givenin [9]. For moduli of smoothness (6.11) and (6.12) the proofs are similar and willbe omitted here (see also [34, 35]). The proof makes use of radial multipliers andbased on boundedness of the translation operator T t .6.3. Properties of the de la Vall´ee Poussin type operators.
Let η ∈S rad ( R d ) be such that η ( x ) = 1 if | x | ≤ , η ( x ) > if | x | < , and η ( x ) = 0 if | x | ≥ . Denote η r ( x ) = 1 − η ( x ) | x | r , b η k,r ( y ) = F k ( η r )( y ) , where F k ( η r ) is a tempered distribution. If t = | x | , η ( t ) = η ( x ) , and η r ( t ) = η r ( x ) , then F k ( η r )( y ) = H λ k ( η r )( | y | ) . Lemma 6.11.
We have b η k,r ∈ L ( R d , dµ k ) , where r > .Proof. It is sufficient to prove that H λ k ( η r ) ∈ L ( R + , dν λ k ) . In the case r ≥ this was proved in [35, (4.25)]. We give the proof for any r > .Letting u j ( t ) = (1 + t ) − j and taking into account that t r = 1(1 + t ) r (cid:16) −
11 + t (cid:17) − r = ∞ X j =0 (cid:18) j + r − j (cid:19) t ) j + r , t = 0 , we obtain, for any M ∈ N and t ≥ , η r ( t ) = ∞ X j =0 (cid:18) j + r − j (cid:19) (1 − η ( t )) u j + r ( t )= M − X j =0 (cid:18) j + r − j (cid:19) u j + r ( t ) − η ( t ) M − X j =0 (cid:18) j + r − j (cid:19) u j + r ( t )+ ∞ X j = M (cid:18) j + r − j (cid:19) (1 − η ( t )) u j + r ( t ) =: ψ ( t ) + ψ ( t ) + ψ ( t ) . Since for any r > we have H λ k ( u r ) ∈ L ( R + , dν λ k ) (see [35, Lemma 3.2], [47,Chapt 5, 5.3.1], [31, Chapt 8, 8.1]), then H λ k ( ψ ) ∈ L ( R + , dν λ k ) . Because of ψ ∈ S ( R + ) , then H λ k ( ψ ) ∈ L ( R + , dν λ k ) . Thus, we are left to show that, forsufficiently large M , H λ k ( ψ ) ∈ L ( R + , dν λ k ) .Let M + r > λ k + 1 , t ≥ . Since Γ( j + r )Γ( j +1) . j r − , we have | ψ ( t ) | ≤ M r − (1 + t ) M + r ∞ X j =0 (1 + j ) r − j . t ) M + r . t M +2 r , p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 29 and Z ∞ | ψ ( t ) | dν λ k ( t ) . Z ∞ t − (2 M +2 r − λ k − dt < ∞ . Thus, ψ ∈ L ( R + , dν λ k ) , H λ k ( ψ ) ∈ C ( R + ) , and H λ k ( ψ ) ∈ L ([0 , , dν λ k ) .Recall that the Bessel differential operator is defined by B λ k = d dt + (2 λ k + 1) t ddt . Using ψ ∈ C ∞ ( R + ) , we have, for any s ∈ N , B sλ k ψ ∈ L ([0 , , dν λ k ) .If t ≥ , then (1 − η ( t )) u j + r ( t ) = u j + r ( t ) and B λ k u j + r ( t ) = 4( j + r )( j + r − λ k ) u j + r +1 ( t ) − j + r )( j + r + 1) u j + r +2 ( t ) . This gives |B λ k u j + r ( t ) | ≤ ( j + r + λ k + 1) u j + r +1 ( t ) . By induction on s , |B sλ k u j + r ( t ) | ≤ s ( j + r + 2 s + λ k − s u j + r + s ( t ) , and then, for t ≥ , |B sλ k ψ ( t ) | . t ) M + r + s ∞ X j =0 (1 + j ) r +2 s − j . t ) M + r + s . t M +2 r +2 s , and B sλ k ψ ∈ L ([2 , ∞ ) , dν λ k ) . Thus, we have B sλ k ψ ∈ L ( R + , dν λ k ) for any s .Choosing s > λ k + 1 and using the inequality |H λ k ( ψ )( τ ) | ≤ τ s Z ∞ |B sλ k ψ ( t ) | dν λ k ( t ) . τ s , we arrive at H λ k ( ψ ) ∈ L ([2 , ∞ ) , dν λ k ) . Finally, we obtain that H λ k ( ψ ) ∈ L ( R + , dν λ k ) . (cid:3) For m, r ∈ N and m ≥ r , we set g ∗ m,r ( y ) := | y | − r j ∗∗ λ k ,m ( | y | ) , g m,r ( x ) := F k ( g ∗ m,r )( x ) ,g tm,r ( x ) := t r − λ k − g m,r (cid:16) xt (cid:17) . Since g ∗ m,r ( y ) = j ∗∗ λ k ,m ( | y | ) η r ( y ) + j ∗∗ λ k ,m ( | y | ) | y | r η ( y ) ,j ∗∗ λ k ,m ( | y | ) | y | r ∈ C ∞ Π ( R d ) , j ∗∗ λ k ,m ( | y | ) | y | r η ( y ) ∈ S ( R d ) , and F k ( j ∗∗ λ k ,m η r )( x ) = (cid:18) mm (cid:19) − m X s = − m ( − s (cid:18) mm − s (cid:19) T s b η λ k ,r ( x ) , then boundedness of the operator T s in L ( R d , dµ k ) and Lemma 6.11 imply that(6.21) g m,r , g tm,r ∈ L ( R d , dµ k ) , k g tm,r k ,dµ k = t r k g m,r k ,dµ k , F − k ( g tm,r )( y ) = F k ( g tm,r )( y ) = t r g ∗ m,r ( ty ) = | y | − r j ∗∗ λ k ,m ( t | y | ) . Lemma 6.12.
Let m, r ∈ N , m ≥ r , ≤ p ≤ ∞ , and f ∈ W rp,k . We have (6.22) ∗∗ ∆ mt f = ( − ∆ k ) r f ∗ k g tm,r and (6.23) k ∗∗ ∆ mt f k p,dµ k . t r k ( − ∆ k ) r f k p,dµ k . Proof.
Let f ∈ S ′ ( R d ) . Applying (6.15), (6.1), (6.4), and (6.21), we obtain F k ( ∗∗ ∆ mt f )( y ) = j ∗∗ λ k ,m ( t | y | ) F k ( f )( y ) = | y | r F k ( f )( y ) j ∗∗ λ k ,m ( t | y | ) | y | r = F k (( − ∆ k ) r f )( y ) F k ( g tm,r )( y ) and (6.22). If f ∈ W rp,k , then ( − ∆ k ) r f ∈ L p ( R d , dµ k ) . Inequality (6.23) followsfrom (6.21), (6.22), Lemma 6.1, and (3.12). Note that a constant in (6.23) can betaken as k g m,r k ,dµ k . (cid:3) Let f ∈ S ′ ( R d ) . We set θ ( x ) = F k ( η )( x ) and θ σ ( x ) = F k ( η ( · /σ ))( x ) . Then θ , θ σ ∈ S ( R d ) . The de la Vall´ee Poussin type operator is given by P σ ( f ) = f ∗ k θ σ .By (6.4),(6.24) F k ( P σ ( f )) = η ( · /σ ) F k ( f )( y ) . Lemma 6.13. If σ > , ≤ p ≤ ∞ , f ∈ L p ( R d , dµ k ) , then (1) k P σ ( f ) k p,dµ k . k f k p,dµ k ; (2) P σ ( f ) ∈ B σp,k and P σ ( g ) = g for any g ∈ B σp,k ; (3) k f − P σ ( f ) k p,dµ k . E σ ( f ) p,dµ k .Remark . Property (3) in this lemma means that P σ ( f ) is the near best ap-proximant of f in L p ( R d , dµ k ) . Proof. (1) Since f ∈ L p ( R d , dµ k ) , η ( · /σ ) , θ σ ∈ S ( R d ) then using Lemma 6.1,(3.12), and the equality k θ k ,dµ k = k θ σ k ,dµ k , we get k P σ ( f ) k p,dµ k = k f ∗ k θ σ k p,dµ k ≤ k θ σ k ,dµ k k f k p,dµ k = k θ k ,dµ k k f k p,dµ k . k f k p,dµ k . (2) We observe that supp η ( · /σ ) ⊂ B σ and then supp F k ( P σ ( f )) ⊂ B σ . Theo-rem 5.14 yields P σ ( f ) ∈ B σp,k . If g ∈ B σp,k , then by Theorem 5.14, supp F k ( g ) ⊂ B σ and F k ( P σ ( g ))( y ) = η ( y/σ ) F k ( g )( y ) = F k ( g )( y ) . Hence, P σ ( g ) = g .(3) Using Theorem 5.15, there exists an entire function g ∗ ∈ B σp,k such that k f − g ∗ k p,dµ k = E σ ( f ) p,dµ k . Then using P σ ( g ∗ ) = g ∗ implies k f − P σ ( f ) k p,dµ k = k f − g ∗ + P σ ( g ∗ − f ) k p,dµ k ≤ k f − g ∗ k p,dµ k + k P σ ( f − g ∗ ) k p,dµ k . E σ ( f ) p,dµ k . (cid:3) In the proof of the next lemma we will use the estimate(6.25) | j ( n ) λ ( t ) | . ( | t | + 1) − ( λ +1 / , t ∈ R , λ ≥ − / , n ∈ Z + , which follows, by induction on n , from the known properties of the Bessel function([2]) | j λ ( t ) | . ( | t | + 1) − ( λ +1 / , j ′ λ ( t ) = − t λ + 1) j λ +1 ( t ) . p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 31 Lemma 6.15. If σ > , ≤ p ≤ ∞ , m ∈ N , r ∈ Z + , f ∈ W rp,k , then (6.26) k f − P σ/ ( f ) k p,dµ k . σ − r k ∗∗ ∆ ma/σ (( − ∆ k ) r f ) k p,dµ k for some a = a ( λ k , m ) > .Proof. If f ∈ S ′ ( R d ) , then by (6.24), (6.15), and (6.1) F k ( f − P σ/ ( f )) = (1 − η (2 · /σ )) F k f = σ − r − η (2 · /σ )( | · | /σ ) r j ∗∗ λ k ,m ( a | · | /σ ) F k ( ∗∗ ∆ ma/σ (( − ∆ k ) r f ))= σ − r ϕ ( · /σ ) F k ( ∗∗ ∆ ma/σ (( − ∆ k ) r f )) , where(6.27) ϕ ( y ) = 1 − η (2 y ) | y | r j ∗∗ λ k ,m ( a | y | ) , ∗∗ ∆ ma/σ (( − ∆ k ) r f ) ∈ L p ( R d , dµ k ) . From here and by (6.4) we have(6.28) f − P σ/ ( f ) = ( ∗∗ ∆ ma/σ (( − ∆ k ) r f ) ∗ k F k ( ϕ ( · /σ ))) . Setting j ∗∗ λ k ,m ( t ) = 1 − τ ( t ) , in light of (6.16) and (6.25), we observe that j ∗∗ λ k ,m ( t ) → as t → ∞ . Then we can choose a > such that | τ ( t ) | ≤ / for | t | ≥ a/ . For such a = a ( λ k , m ) , we have that ϕ ( y ) = 0 for | y | ≤ / , ϕ ( y ) > for | y | > / , and ϕ ∈ C ∞ Π ( R d ) .We will use the following decomposition ϕ ( y ) = ϕ ( | y | ) + ϕ ( | y | ) , where ϕ ( | y | ) = 2 r η r (2 y ) (cid:16) − τ ( a | y | ) − S N ( τ ( a | y | ) (cid:17) and ϕ ( | y | ) = 2 r η r (2 y ) S N ( τ ( a | y | )) , η r ( y ) = 1 − η ( y ) | y | r , S N ( t ) = N − X j =0 t j . First, we show that F k ( ϕ ( | · | )) ∈ L ( R d , dµ k ) . Since for a radial function wehave ∆ k ϕ ( | y | ) = ϕ ′′ ( | y | ) + 2 λ k + 1 | y | ϕ ′ ( | y | ) and, for | t | ≤ / , (1 − t ) − − N − X j =0 t j = (1 − t ) − − S N ( t ) = t N − t , then, by (6.16) and (6.25), we obtain ∆ sk ϕ ( | y | ) = O ( | y | − r − N ( λ k +1 / ) , | y | ≥ / , s ∈ Z + . Hence, for a fixed N ≥ / (2 λ k + 1) , we have ∆ sk ϕ ( | y | ) ∈ L ( R d , dµ k ) , where s ∈ Z + . Since F k (( − ∆ k ) s ϕ ( | · | ))( x ) = | x | s F k ( ϕ ( | · | ))( x ) , then |F k ( ϕ ( | · | ))( x ) | = | x | − s (cid:12)(cid:12)Z R d e k ( − x, y )( − ∆ k ) s ϕ ( | y | ) dµ k ( y ) (cid:12)(cid:12) ≤ k ( − ∆ k ) s ϕ ( | y | ) k ,dµ k | x | s . Setting s > λ k + 1 yields F k ( ϕ ( | · | )) ∈ L ( R d , dµ k ) .Second, let us show that F k ( ϕ ( | · | )) ∈ L ( R d , dµ k ) for r ∈ N . Let τ ( t ) = m X s =1 ν s j λ k ( st ) , ψ r ( x ) = 2 r F k ( η r (2 · ))( x ) ,A a f ( x ) = m X s =1 ν s T as f ( x ) , B a f ( x ) = N − X j =0 ( A a ) j f ( x ) . Boundedness of the operator T t in L p ( R d , dµ k ) implies k A a k p → p = sup {k Af k p,dµ k : k f k p,dµ k ≤ } ≤ m X s =1 | ν s | and(6.29) k B a k p → p ≤ N − X j =0 ( k A k p → p ) j ≤ N (cid:16) m X s =1 | ν s | (cid:17) N − , ≤ p < ∞ . Then for p = 1 , taking into account Lemma 6.11, we have kF k ( ϕ ( | · | )) k ,dµ k = (cid:13)(cid:13) B a ψ r (cid:13)(cid:13) ,dµ k ≤ N (cid:16) m X s =1 | ν s | (cid:17) N − k ψ r k ,dµ k < ∞ . Thus, F k ( ϕ ) ∈ L ( R d , dµ k ) . Combining Lemma 6.1, relations (3.12), (6.28), (6.27),and the formula kF k (( · /σ )) k ,dµ k = kF k ( ϕ ) k ,dµ k , we obtain inequality (6.26) for r ∈ N .Let now r = 0 . Define the operators A and A as follows: F k ( A g )( y ) = ϕ ( | y | /σ ) F k ( g )( y ) and F k ( A g )( y ) = ϕ ( | y | /σ ) F k ( g )( y ) , ϕ ( | y | ) = (1 − η (2 y )) S N ( τ ( a | y | )) . Since F k ( ϕ ( | · | )) ∈ L ( R d , dµ k ) , then by (3.12) for ≤ p ≤ ∞ (6.30) k A g k p,dµ k ≤ kF k ( ϕ ( | y | )) k ,dµ k k g k p,dµ k . k g k p,dµ k , g ∈ L p ( R d , dµ k ) . We are left to show that k A g k p,dµ k . k g k p,dµ k , ≤ p ≤ ∞ , g ∈ L p ( R d , dµ k ) . We have F k ( A g )( y ) = (1 − η (2 y/σ )) S N ( τ ( a | y | /σ )) F k ( g )( y )= (1 − η (2 y/σ )) F k ( B a/σ g )( y )= F k ( B a/σ g − P σ/ ( B a/σ g ))( y ) . Since B a/σ g ∈ L p ( R d , dµ k ) , using Lemma 6.13 and inequality (6.29), we get(6.31) k A g k p,dµ k . k B a/σ g k p,dµ k ≤ N (cid:0) m X s =1 | ν s | (cid:1) N − k g k p,dµ k . k g k p,dµ k . Using (6.30) and (6.31) with g = ∗∗ ∆ ma/σ f , we finally obtain (6.26) for r = 0 . (cid:3) p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 33 Lemma 6.16. If σ > , ≤ p ≤ ∞ , m ∈ N , f ∈ L p ( R d , dµ k ) , then (6.32) k (( − ∆ k ) m P σ ( f ) k p,dµ k . σ m k ∗∗ ∆ ma/ (2 σ ) f k p,dµ k , where a = a ( λ k , m ) > is given in Lemma 6.15.Proof. We have F k ((( − ∆ k ) m P σ ( f ))( y ) = | y | m η ( y/σ ) F k ( f )( y )= σ m ϕ ( y/σ ) j ∗∗ λ k ,m ( a/ (2 σ )) F k ( f )( y )= σ m ϕ ( y/σ ) F k ( ∗∗ ∆ ma/ (2 σ ) f )( y ) , where ϕ ( y ) = | y | m η ( y ) j ∗∗ λ k ,m ( a | y | / . Since j ∗∗ λ k ,m ( a | y | / / | y | m > for | y | > , we observe that ϕ ∈ S ( R d ) and F k ( ϕ ) ∈ L ( R d , dµ k ) . Then estimate (6.32) follows from Lemma 6.1, (3.12), ∗∗ ∆ ma/ (2 σ ) f ∈ L p ( R d , dµ k ) , and kF k ( ϕ ( · /σ )) k ,dµ k = kF k ( ϕ ) k ,dµ k . (cid:3) Proofs of Theorem 6.6 and 6.8.
Proof of Theorem 6.8.
In connection with Lemma 6.12, observe that, for f ∈ L p ( R d , dµ k ) and g ∈ W rp,k , k ∗∗ ∆ rδ f k p,dµ k ≤ ∗∗ ω r ( δ, f ) p,dµ k ≤ ∗∗ ω r ( δ, f − g ) p,dµ k + ∗∗ ω r ( δ, g ) p,dµ k . ( k f − g k p,dµ k + δ r k ( − ∆ k ) r g k p,dµ k ) . Then(6.33) k ∗∗ ∆ rδ f k p,dµ k ≤ ∗∗ ω r ( δ, f ) p,dµ k . K r ( δ, f ) p,dµ k . On the other hand, P σ ( f ) ∈ W rp,k and(6.34) K r ( δ, f ) p,dµ k ≤ k f − P σ ( f ) k p,dµ k + δ r k ( − ∆ k ) r P σ ( f ) k p,dµ k . In light of Lemma 6.15, k f − P σ ( f ) k p,dµ k . k ∗∗ ∆ ra/ (2 σ ) f k p,dµ k . Further, Lemma 6.16 yields(6.35) k (( − ∆ k ) r P σ ( f ) k p,dµ k . σ r k ∗∗ ∆ ra/ (2 σ ) f k p,dµ k . Setting σ = a/ (2 δ ) , from (6.34)–(6.35) we arrive at(6.36) K r ( δ, f ) p,dµ k . k ∗∗ ∆ rδ f k p,dµ k . ∗∗ ω r ( δ, f ) p,dµ k . (cid:3) Proof of Theorem 6.6.
Using property (6.7), inequalities (6.33) and (6.36), we ob-tain E σ ( f ) p,dµ k ≤ k f − P σ/ ( f ) k p,dµ k . σ − r k ∗∗ ∆ ma/σ (( − ∆ k ) r f ) k p,dµ k . σ r K m (cid:16) aσ , ( − ∆ k ) r f (cid:17) p,dµ k . σ r K m (cid:16) σ , ( − ∆ k ) r f (cid:17) p,dµ k . σ r k ∗∗ ∆ m /σ (( − ∆ k ) r f ) k p,dµ k . σ r ∗∗ ω m (cid:16) σ , ( − ∆ k ) r f (cid:17) p,dµ k . (6.37) (cid:3) Remark . The proofs of estimates (6.36) and (6.37) for the difference (6.10) isbased on the fact that the parameter a in Lemmas 6.15 and 6.16 is the same. It ispossible due to the fact that j ∗∗ λ k ,m ( t ) > for t > , see Remark 6.4. This estimateis valid for the difference (6.8) as well, since j λ k ,m ( t ) = (1 − j λ k ( t )) m > for t > .Therefore, the moduli of smoothness (6.11) and (6.13) in inequalities (6.19) and(6.20) can be replaces by the norms of the corresponding differences (6.8) and(6.10). For the modulus of smoothness (6.12) this observation is not valid since j ∗ λ k ,m ( t ) does not keep its sign. Remark . Properties (6.6) and (6.7) of the K -functional, inequality (6.18) andthe equivalence (6.20) imply the following properties of moduli of smoothness(1) lim δ → ω m ( δ, f ) p,dµ k = lim δ → ∗ ω m ( δ, f ) p,dµ k = lim δ → ∗∗ ω m ( δ, f ) p,dµ k = 0; (2) ω m ( λδ, f ) p,dµ k . max { , λ m } ω m ( δ, f ) p,dµ k ; (3) ∗ ω l ( λδ, f ) p,dµ k . max { , λ m } ∗ ω l ( δ, f ) p,dµ k , l = 2 m − , m ; (4) ∗∗ ω m ( λδ, f ) p,dµ k . max { , λ m } ∗∗ ω m ( δ, f ) p,dµ k ; (5) ∗ ω m + r ( δ, f ) p,dµ k . ∗ ω m ( δ, f ) p,dµ k ; (6) ∗∗ ω m + r ( δ, f ) p,dµ k . ∗∗ ω m ( δ, f ) p,dµ k . Some inequalities for entire functions
In this section, we study weighted analogues of the inequalities for entire func-tions. In particular, we obtain Nikolskii’s inequality ([31], see Theorem 7.1 be-low), Bernstein’s inequality ([31], Theorem 7.3), Nikolskii–Stechkin’s inequality([30, 45], Theorem 7.5), and Boas-type inequality ([4], Theorem 7.7).
Theorem 7.1. If σ > , < p ≤ q ≤ ∞ , f ∈ B σp,k , then (7.1) k f k q,dµ k . σ (2 λ k +2)(1 /p − /q ) k f k p,dµ k . Remark . Observe that the obtained Nikolskii inequality is sharp, i.e., we ac-tually have sup f ∈ B σp,k ,f =0 k f k q,dµ k k f k p,dµ k ≍ σ (2 λ k +2)(1 /p − /q ) , and an extremizer can be taken as f σ,m ( x ) = sin m ( θ | x | ) | x | m , θ = σ m , for sufficiently large m ∈ N . Proof.
Let f ∈ B σp,k , p ≥ , q = ∞ . By Theorem 5.14, we have supp F k ( f ) ⊂ B σ ,and then(7.2) F k ( f )( y ) = η ( y/σ ) F k ( f )( y ) , η ( y ) = η ( | y | ) . Lemma 3.9 implies f ( x ) = ( f ∗ λk H λ k ( η ( · /σ )))( x ) = Z ∞ T t f ( x ) H λ k ( η ( · /σ ))( t ) dν λ k ( t ) . Taking into account that H λ k ( η ( · /σ ))( t ) = σ λ k +2 H λ k ( η )( σt ) , p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 35 kH λ k ( η )( σt ) k p ′ ,dµ k = σ − λk +2 p ′ kH λ k ( η )( t ) k p ′ ,dµ k , H¨older’s inequality and Theorem 3.5 yield | f ( x ) | ≤ σ λ k +2 k T t f ( x ) k p,dν λk kH λ k ( η )( σt ) k p ′ ,dµ k ≤ σ (2 λ k +2) /p kH λ k ( η )( t ) k p ′ ,dµ k k f k p,dµ k . σ (2 λ k +2) /p k f k p,dµ k , i.e., (7.1) holds.Let f ∈ B σp,k , < p < , q = ∞ . By Theorem 5.1, f is bounded and f ∈ B σ ,k .We have k f k ,dµ k = k| f | − p | f | p k ,dµ k ≤ k| f | − p k ∞ k| f | p k ,dµ k = k f k − p ∞ k f k pp,dµ k . Using (7.1) with p = 1 and q = ∞ , k f k ,dµ k . σ λ k +2 k f k ,dµ k k f k − p ∞ k f k pp,dµ k , which gives k f k ∞ . σ (2 λ k +2) /p k f k p,dµ k . Thus, the proof of (7.1) for q = ∞ is complete.If < p ≤ q < ∞ , we obtain k f k q,dµ k = k| f | − p/q | f | p/q k q,dµ k ≤ k f k − p/q ∞ k f k p/qp,dµ k ≤ σ (2 λ k +2)(1 − p/q ) /p k f k − p/qp,dµ k k f k p/qp,dµ k = σ (2 λ k +2)(1 /p − /q ) k f k p,dµ k . (cid:3) Theorem 7.3. If σ > , r ∈ N , ≤ p ≤ ∞ , f ∈ B σp,k , then (7.3) k ( − ∆ k ) r f k p,dµ k . σ r k f k p,dµ k . Proof.
It is enough to consider the case r = 1 . As in the previous theorem, we use(7.2) to obtain F k (( − ∆ k ) f )( y ) = | y | η ( y/σ ) F k ( f )( y ) = σ ϕ ( | y | /σ ) F k ( f )( y ) , where ϕ ( t ) = t η ( t ) ∈ S ( R + ) . Combining Lemma 3.9, inequality (3.12), and kF k ( ϕ ( | · | /σ )) k ,dµ k = kF k ( ϕ ( | · | )) k ,dµ k , we arrive at k ( − ∆ k ) f k p,dµ k ≤ σ kF k ( ϕ ( | · | )) k ,dµ k k f k p,dµ k . σ k f k p,dµ k . (cid:3) The next result follows from Lemma 6.12, and Theorem 7.3.
Corollary 7.4. If σ, δ > , m ∈ N , ≤ p ≤ ∞ , f ∈ B σp,k , then ω m ( δ, f ) p,dµ k . ( σδ ) m k f k p,dµ k , ∗ ω l ( δ, f ) p,dµ k . ( σδ ) m k f k p,dµ k , l = 2 m − , m, ∗∗ ω m ( δ, f ) p,dµ k . ( σδ ) m k f k p,dµ k , where constants do not depend on σ, δ, and f . Theorem 7.5. If σ > , m ∈ N , ≤ p ≤ ∞ , < t ≤ / (2 σ ) , f ∈ B σp,k , then (7.4) k ( − ∆ k ) m f k p,dµ k . t − m k ∗∗ ∆ mt f k p,dµ k . Remark . By Remark 6.10, this inequality can be equivalently written as k ( − ∆ k ) m f k p,dµ k . t − m K m ( t, f ) p,dµ k . Proof.
We have F k (( − ∆ k ) m f )( y ) = | y | m η ( y/σ ) j ∗∗ λ k ,m ( t | y | ) j ∗∗ λ k ,m ( t | y | ) F k ( f )( y ) . Since for < t ≤ / (2 σ ) η ( y/σ ) = η ( y/σ ) η ( ty ) , we obtain that F k (( − ∆ k ) m f )( y ) = t − m η ( y/σ ) ϕ ( ty ) j ∗∗ λ k ,m ( t | y | ) F k ( f )( y ) , where ϕ ( y ) = | y | m η ( y ) j ∗∗ λ k ,m ( | y | ) ∈ S ( R d ) . Using j ∗∗ λ k ,m ( t | · | ) F k ( f ) = F k ( ∗∗ ∆ mt f ) , ∗∗ ∆ mt f ∈ L p ( R d , dµ k ) , and kF k ( η ( · /σ )) k ,dµ k = kF k ( η ) k ,dµ k , kF k ( ϕ ( t · )) k ,dµ k = kF k ( ϕ ) k ,dµ k , and combining Lemma 3.9 and inequality (3.12), we have k ( − ∆ k ) m f k p,dµ k ≤ t − m kF k ( η ( · /σ )) k ,dµ k kF k ( ϕ ( t · )) k ,dµ k k ∗∗ ∆ mt f k p,dµ k = t − m kF k ( η ) k ,dµ k kF k ( ϕ ) k ,dµ k k ∗∗ ∆ mt f k p,dµ k . t − m k ∗∗ ∆ mt f k p,dµ k . (cid:3) Theorem 7.7. If σ > , m ∈ N , ≤ p ≤ ∞ , < δ ≤ t ≤ / (2 σ ) , f ∈ B σp,k , then (7.5) δ − m k ∗∗ ∆ mδ f k p,dµ k . t − m k ∗∗ ∆ mt f k p,dµ k . Remark . Using Remark 6.10, Theorem 7.5, and taking into account that δ − m K m ( δ, f ) p,dµ k is decreasing in δ (see (6.7)), inequality (7.5) can be equiv-alently written as k ( − ∆ k ) m f k p,dµ k ≍ δ − m k ∗∗ ∆ mδ f k p,dµ k ≍ t − m k ∗∗ ∆ mt f k p,dµ k , k ( − ∆ k ) m f k p,dµ k ≍ δ − m K m ( δ, f ) p,dµ k ≍ t − m K m ( t, f ) p,dµ k . Proof.
We have F k ( ∗∗ ∆ mδ f )( y ) = j ∗∗ λ k ,m ( δ | y | ) F k ( f )( y )= η ( y/σ ) j ∗∗ λ k ,m ( δ | y | ) η ( ty ) j ∗∗ λ k ,m ( t | y | ) F k ( ∗∗ ∆ mt f )( y )= θ m η ( y/σ ) ϕ θ ( ty ) F k ( ∗∗ ∆ mt f )( y ) , where θ = δ/t ∈ (0 , , ϕ θ ( y ) = ψ ( θy ) η ( y ) ψ ( y ) ∈ S ( R d ) , ψ ( y ) = j ∗∗ λ k ,m ( | y | ) | y | m ∈ C ∞ ( R d ) . Using Lemma 3.9 and estimate (3.12), we arrive at inequality (7.5): k ∗∗ ∆ mδ f k p,dµ k ≤ θ m kF k ( η ) k ,dµ k max ≤ θ ≤ kF k ( ϕ θ ) k ,dµ k k ∗∗ ∆ mt f k p,dµ k . (cid:16) δt (cid:17) m k ∗∗ ∆ mt f k p,dµ k , provided that the function n ( θ ) = kF k ( ϕ θ ) k ,dµ k is continuous on [0 , . Let usprove this. p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 37 Set ϕ θ ( y ) = ϕ θ ( | y | ) , r = | y | , ρ = | x | . Then n ( θ ) = Z R d (cid:12)(cid:12)(cid:12)Z R d ϕ θ ( y ) e k ( x, y ) dµ k ( y ) (cid:12)(cid:12)(cid:12) dµ k ( x )= Z ∞ (cid:12)(cid:12)(cid:12)Z ϕ θ ( r ) j λ k ( ρr ) dν λ k ( r ) (cid:12)(cid:12)(cid:12) dν λ k ( ρ )= b λ k Z ∞ (cid:12)(cid:12)(cid:12)Z ϕ θ ( r ) j λ k ( ρr ) r λ k +1 dr (cid:12)(cid:12)(cid:12) ρ λ k +1 dρ. The inner integral continuously depends on θ . Let us show that the outer integralconverges uniformly in θ ∈ [0 , . Since [2, Sect. 7.2] ddr (cid:0) j λ k +1 ( ρr ) r λ k +2 (cid:1) = (2 λ k + 2) j λ k ( ρr ) r λ k +1 , integrating by parts implies Z ϕ θ ( r ) j λ k ( ρr ) r λ k +1 dr = Z ϕ θ ( r ) d (cid:16)Z r j λ k ( ρτ ) τ λ k +1 (cid:17) = − λ k + 2 Z ( ϕ θ ( r )) ′ r j λ k +1 ( ρr ) r λ k +3 dr = . . . = ( − s (cid:16) s Y j =1 (2 λ k + 2 s ) (cid:17) − Z ϕ [ s ] θ ( r ) j λ k + s ( ρr ) r λ k +2 s +1 dr, where ϕ [ s ] θ ( r ) := ddr ϕ [ s − θ ( r ) r ∈ C ∞ ( R + × [0 , , since ϕ θ ( r ) is even in r and ϕ θ ∈ C ∞ ( R + × [0 , . This and (6.25) give (cid:12)(cid:12)(cid:12)Z ϕ θ ( r ) j λ k ( ρr ) r λ k +1 dr (cid:12)(cid:12)(cid:12) ≤ c ( λ k , m, s )( ρ + 1) λ k + s +1 / and, for s > λ k + 3 / , n ( θ ) ≤ c ( λ k , m, s ) Z ∞ (1 + ρ ) − ( s − λ k − / dρ ≤ c ( λ k , m, s ) , completing the proof. (cid:3) Remark . Combining (7.1) and (7.3), the following Bernstein–Nikolskii inequal-ity is valid k ( − ∆ k ) r f k q,dµ k . σ r +(2 λ k +2)(1 /p − /q ) k f k p,dµ k , ≤ p ≤ q ≤ ∞ . Remark . For radial functions, Nikolskii inequality (7.1), Bernstein (7.3),Nikolskii–Stechkin (7.4), and Boas inequality (7.5) follow from corresponding es-timates in the space L p ( R + , dν λ ) proved in [34]. Realization of K -functionals and moduli of smoothness In the non-weighted case ( k ≡ ) the equivalence between the classical modulusof smoothness and the K -functional between L p and the Sobolev space W rp is wellknown [8, 26]: ≤ p ≤ ∞ , for any integer r one has ω r ( t, f ) L p ( R ) ≍ K r ( f, t ) p , ≤ p ≤ ∞ , where K r ( f, t ) p := inf g ∈ ˙ W rp (cid:0) k f − g k p + t r k g k ˙ W rp (cid:1) . Starting from the paper [13] (see also [17, Lemma 1.1] for the fractional case),the following equivalence between the modulus of smoothness and the realizationof the K -functional is widely used in approximation theory: ω r ( t, f ) L p ( R ) ≍ R r ( t, f ) p = inf g {k f − g k p + t r k g ( r ) k p } , where g is an entire function of exponential type /t. Let the realization of the K -functional K r ( t, f ) p,dµ k be given as follows: R r ( t, f ) p,dµ k = inf {k f − g k p,dµ k + t r k ( − ∆ k ) r g k p,dµ k : g ∈ B /tp,k } and R ∗ r ( t, f ) p,dµ k = k f − g ∗ k p,dµ k + t r k ( − ∆ k ) r g ∗ k p,dµ k , where g ∗ ∈ B /tp,k is a near best approximant. Theorem 8.1. If t > , ≤ p ≤ ∞ , r ∈ N , then for any f ∈ L p ( R d , dµ k ) R r ( t, f ) p,dµ k ≍ R ∗ r ( t, f ) p,dµ k ≍ K r ( t, f ) p,dµ k ≍ ω r ( t, f ) p,dµ k ≍ ∗∗ ω r ( t, f ) p,dµ k ≍ ∗ ω r − ( t, f ) p,dµ k ≍ ∗ ω r ( t, f ) p,dµ k . Proof.
By Theorem 6.8, ω r ( t, f ) p,dµ k ≍ ∗∗ ω r ( t, f ) p,dµ k ≍ ∗ ω r − ( t, f ) p,dµ k ≍ ∗ ω r ( t, f ) p,dµ k ≍ K r ( t, f ) p,dµ k ≤ R r ( t, f ) p,dµ k ≤ R ∗ r ( t, f ) p,dµ k , where we have used the fact that B /tp,k ⊂ W rp,k , which follows from Theorem 7.3.Therefore, it is enough to show that R ∗ r ( t, f ) p,dµ k ≤ Cω r ( t, f ) p,dµ k . Indeed, for g ∗ being the best approximant (or near best approximant), the Jacksoninequality given in Theorem 6.6 implies that(8.1) k f − g ∗ k p,dµ k . E /t ( f ) p,dµ k . ω r ( t, f ) p,dµ k . Using the first inequality in Theorem 7.5 and taking into account (8.1), we have k ( − ∆ k ) r g ∗ k p,dµ k . t − r k ∆ rt/ g ∗ k p,dµ k . t − r k ∆ rt/ ( g ∗ − f ) k p,dµ k + t − r k ∆ rt/ f k p,dµ k . t − r k g ∗ − f k p,dµ k + t − r ω r ( t/ , f ) p,dµ k . Using again (8.1), we arrive at k f − g ∗ k p,dµ k + t r k ( − ∆ k ) r g ∗ k p,dµ k . ω r ( t, f ) p,dµ k , completing the proof. (cid:3) p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 39 The next result answers the following important question (see, e.g., [51, 22]):when does the relation(8.2) ω m (cid:16) n , f (cid:17) p,dµ k ≍ E n ( f ) p,dµ k (or similar relations with concepts in Theorem 8.2) hold? Theorem 8.2.
Let ≤ p ≤ ∞ and m ∈ N . We have that (8.2) is valid if andonly if (8.3) ω m (cid:16) n , f (cid:17) p,dµ k ≍ ω m +1 (cid:16) n , f (cid:17) p,dµ k . Proof.
We prove only the non-trivial part that (8.3) implies (8.2). Another part ofthe Theorem 8.2 follows from (6.18), Jackson’s inequality (6.19) and Remark 6.18.Since, by (6.7) we have ω m ( nt, f ) p,dµ k . n m ω m ( t, f ) p,dµ k , relation (8.3) impliesthat(8.4) ω m +1 ( nt, f ) p,dµ k . n m ω m +1 ( t, f ) p,dµ k . This and Jackson’s inequality give n m +1) n X j =0 ( j + 1) m +1) − E j ( f ) p,dµ k . n m +1) n X j =0 ( j + 1) m +1) − ω m +1 (cid:16) j + 1 , f (cid:17) p,dµ k . ω m +1 (cid:16) n , f (cid:17) p,dµ k . Moreover, Theorem 9.1 below implies ω m +1 (cid:16) ln , f (cid:17) p,dµ k . ln ) m +1) ln X j =0 ( j + 1) m +1) − E j ( f ) p,dµ k . l m +1) ω m +1 (cid:16) n , f (cid:17) p,dµ k + 1( ln ) m +1) ln X j = n +1 ( j + 1) m +1) − E j ( f ) p,dµ k , or, in other words, n m +1) ln X j = n +1 ( j + 1) m +1) − E j ( f ) p,dµ k & Cl m +1) ω m +1 (cid:16) ln , f (cid:17) p,dµ k − ω m +1 (cid:16) n , f (cid:17) p,dµ k . Using again (8.4), we obtain n m +1) ln X j = n +1 ( j + 1) m +1) − E j ( f ) p,dµ k & ( Cl − ω m +1 (cid:16) n , f (cid:17) p,dµ k . Taking into account monotonicity of E j ( f ) p,dµ k and choosing l sufficiently large,we arrive at (8.2). (cid:3) Inverse theorems of approximation theory
Theorem 9.1.
Let m, n ∈ N , ≤ p ≤ ∞ , f ∈ L p ( R d , dµ k ) . We have (9.1) K m (cid:16) n , f (cid:17) p,dµ k . n m n X j =0 ( j + 1) m − E j ( f ) p,dµ k . Remark . By Remark 6.10, K m (cid:0) n , f (cid:1) p,dµ k in this inequality can be equivalentlyreplaced by ω m (cid:0) n , f (cid:1) p,dµ k , ∗∗ ω m (cid:0) n , f (cid:1) p,dµ k , and ∗ ω l (cid:0) n , f (cid:1) p,dµ k , l = 2 m − , m . Proof.
Let us prove (9.1) for ω m (cid:0) n , f (cid:1) p,dµ k . By Theorem 5.15, for any σ > thereexists f σ ∈ B σp,k such that k f − f σ k p,dµ k = E σ ( f ) p,dµ k , E ( f ) p,dµ k = k f k p,dµ k . For any s ∈ Z + , ω m (1 /n, f ) p,dµ k ≤ ω m (1 /n, f − f s +1 ) p,dµ k + ω m (1 /n, f s +1 ) p,dµ k . E s +1 ( f ) p,dµ k + ω m (1 /n, f s +1 ) p,dµ k . Using Lemma 6.12, ω m (1 /n, f s +1 ) p,dµ k . n − m k ( − ∆ k ) m f s +1 k p,dµ k . n m (cid:16) k ( − ∆ k ) m f k p,dµ k + s X j =0 k ( − ∆ k ) m f j +1 − ( − ∆ k ) m f j k p,dµ k (cid:17) . Then Bernstein inequality (7.3) implies that k ( − ∆ k ) m f j +1 − ( − ∆ k ) m f j k p,dµ k . m ( j +1) k f j +1 − f j k p,dµ k . m ( j +1) E j ( f ) p,dµ k , k ( − ∆ k ) m f k p,dµ k . E ( f ) p,dµ k . Thus, ω m (1 /n, f s +1 ) p,dµ k . n m (cid:16) E ( f ) p,dµ k + s X j =0 m ( j +1) E j ( f ) p,dµ k (cid:17) . Taking into account that(9.2) j X l =2 j − +1 l m − E l ( f ) p,dµ k ≥ m ( j − E j ( f ) p,dµ k , we have ω m (1 /n, f s +1 ) p,dµ k . n m (cid:16) E ( f ) p,dµ k + 2 m E ( f ) p,dµ k + s X j =1 m j X l =2 j − +1 l m − E l ( f ) p,dµ k (cid:17) . n m s X j =0 ( j + 1) m − E j ( f ) p,dµ k . Choosing s such that s ≤ n < s +1 implies (9.1). (cid:3) Theorem 9.1 and Jackson’s inequality imply the following Marchaud inequality.
Corollary 9.3.
Let m ∈ N , ≤ p ≤ ∞ , f ∈ L p ( R d , dµ k ) . We have K m ( δ, f ) p,dµ k . δ m (cid:16) k f k p,dµ k + Z δ t − m K m +2 ( t, f ) p,dµ k dtt (cid:17) . p -BOUNDED DUNKL-TYPE GENERALIZED TRANSLATION OPERATOR 41 Theorem 9.4.
Let ≤ p ≤ ∞ , f ∈ L p ( R d , dµ k ) and r ∈ N be such that P ∞ j =1 j r − E j ( f ) p,dµ k < ∞ . Then f ∈ W rp,k and, for any m, n ∈ N , we have (9.3) K m (cid:16) n , ( − ∆ k ) r f (cid:17) p,dµ k . n r n X j =0 ( j + 1) k +2 r − E j ( f ) p,dµ k + ∞ X j = n +1 j r − E j ( f ) p,dµ k . Remark . We can replace K m (cid:0) n , ( − ∆ k ) r f (cid:1) p,dµ k by any of moduli ω m (cid:0) n , ( − ∆ k ) r f (cid:1) p,dµ k , ∗ ω l (cid:0) n , ( − ∆ k ) r f (cid:1) p,dµ k , and ∗∗ ω m (cid:0) n , ( − ∆ k ) r f (cid:1) p,dµ k , l = 2 m − , m . Proof.
Let us prove (9.3) for ω m (cid:0) n , ( − ∆ k ) r f (cid:1) p,dµ k . Consider(9.4) ( − ∆ k ) r f + ∞ X j =0 (cid:0) ( − ∆ k ) r f j +1 − ( − ∆ k ) r f j (cid:1) . By Bernstein’s inequality (7.3), k ( − ∆ k ) r f j +1 − ( − ∆ k ) r f j k p,dµ k . ( j +1) r E j ( f ) p,dµ k . j X l =2 j − +1 l r − E l ( f ) p,dµ k . Therefore, series (9.4) converges to a function g ∈ L p ( R d , dµ k ) . Let us show that g = ( − ∆ k ) r f , i.e., f ∈ W rp,k . Set S N = ( − ∆ k ) r f + N X j =0 (cid:0) ( − ∆ k ) r f j +1 − ( − ∆ k ) r f j (cid:1) . Then hF k ( g ) , ϕ i = h g, F k ( ϕ ) i = lim N →∞ h S N , F k ( ϕ ) i == lim N →∞ hF k ( S N ) , ϕ i = lim N →∞ h| y | r F k ( f N +1 ) , ϕ i = h| y | r F k ( f ) , ϕ i , where ϕ ∈ S ( R d ) . Hence, F k ( g )( y ) = | y | r F k ( f )( y ) and g = ( − ∆ k ) r f .To obtain (9.3), we write ω m (1 /n, ( − ∆ k ) r f ) p,dµ k ≤ ω m (1 /n, ( − ∆ k ) r f − S N ) p,dµ k + ω m (1 /n, S N ) p,dµ k . The first term is estimated as follows ω m (1 /n, ( − ∆ k ) r f − S N ) p,dµ k . k ( − ∆ k ) r f − S N k p,dµ k . ∞ X j = N +1 r ( j +1) E j ( f ) p,dµ k . ∞ X l =2 N +1 l r − E l ( f ) p,dµ k . Moreover, by Corollary 7.4, ω m (1 /n, S N ) p,dµ k ≤ ω m (1 /n, ( − ∆ k ) r f ) p,dµ k + N X j =0 ω m (cid:0) /n, ( − ∆ k ) r f j +1 − ( − ∆ k ) r f j (cid:1) p,dµ k . n r (cid:16) E ( f ) p,dµ k + N X j =0 m + r )( j +1) E j ( f ) p,dµ k (cid:17) . Using (9.2) and choosing N such that N ≤ n < N +1 completes the proof of(9.3). (cid:3) References [1] J.-P. Anker,
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E-mail address : [email protected] V. Ivanov, Tula State University, Department of Applied Mathematics andComputer Science, 300012 Tula, Russia
E-mail address : [email protected] S. Tikhonov, ICREA, Centre de Recerca Matem`atica, and UAB, Campus deBellaterra, Edifici C 08193 Bellaterra (Barcelona), Spain
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