We study the dual Dunkl-Sonine operator tSk,e on Rd and give expression of tSk,t, using Dunkl multiplier operators on Rd, Next, we study the extremal functions fλ, λ〉 0 related to the Dunkl multiplier operators, an...We study the dual Dunkl-Sonine operator tSk,e on Rd and give expression of tSk,t, using Dunkl multiplier operators on Rd, Next, we study the extremal functions fλ, λ〉 0 related to the Dunkl multiplier operators, and more precisely show that {fλ}λ〉0 converges uniformly to tSk,e（f） as λ→0 Certain examples based on Dunkl-heat and Dunkt-Poisson kernels are provided to illustrate the results.展开更多
Abstract The aim of this paper is to prove duality and reflexivity of generalized Lipschitz spaces ∧κα,p,q（R）, α ∈R and 1≤〈 p, q ≤∞, in the context of Dunkl harmonic analysis.
In this paper, we study the sharp Jackson inequality for the best approximation of f ∈L2,k（Rd） by a subspace Ek2（σ） （SEk2（σ））, which is a subspace of entire functions of exponential type （spherical exponen...In this paper, we study the sharp Jackson inequality for the best approximation of f ∈L2,k（Rd） by a subspace Ek2（σ） （SEk2（σ））, which is a subspace of entire functions of exponential type （spherical exponential type） at most σ. Here L2,k（Rd） denotes the space of all d-variate functions f endowed with the L2-norm with the weight vk（x）=Пζ∈R＋}（ζ,x）}2k（ζ）,which is defined by a positive subsystem R＋ of a finite root system R Rd and a function k（ζ）：R→R＋ invariant under the reflection group G（R） generated by R. In the case G（R） = Z2d, we get some exact results. Moreover, the deviation of best approximation by the subspace Ek2（σ） （SE2（σ）） of some class of the smooth functions in the space L2,k（Rd） is obtained.展开更多
We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator D in the upper half-plane R2+=R×(0,∞),where(Df)(x)=f′0(x)+(λ/x)[f(x)-f(-x)]for givenλ≥0.A C...We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator D in the upper half-plane R2+=R×(0,∞),where(Df)(x)=f′0(x)+(λ/x)[f(x)-f(-x)]for givenλ≥0.A C2 function u in R2+is said to beλ-harmonic if(D2x+■2y)u=0.For aλ-harmonic function u in R2+and for a subset E of■R2+=R symmetric about y-axis,we prove that the following three assertions are equivalent:(i)u has a finite non-tangential limit at(x,0)for a.e.x∈E;(ii)u is non-tangentially bounded for a.e.x∈E;(iii)(Su)(x)<∞for a.e.x∈E,where S is a Lusin-type area integral associated with the Dunkl operator D.展开更多
基金partially supported by DGRST project04/UR/15-02CMCU program 10G 1503
文摘We study the dual Dunkl-Sonine operator tSk,e on Rd and give expression of tSk,t, using Dunkl multiplier operators on Rd, Next, we study the extremal functions fλ, λ〉 0 related to the Dunkl multiplier operators, and more precisely show that {fλ}λ〉0 converges uniformly to tSk,e（f） as λ→0 Certain examples based on Dunkl-heat and Dunkt-Poisson kernels are provided to illustrate the results.
文摘Abstract The aim of this paper is to prove duality and reflexivity of generalized Lipschitz spaces ∧κα,p,q（R）, α ∈R and 1≤〈 p, q ≤∞, in the context of Dunkl harmonic analysis.
基金Supported by National Natural Science Foundation of China(Grant No.11071019)the research Fund for the Doctoral Program of Higher Education and Beijing Natural Science Foundation(Grant No.1102011)
文摘In this paper, we study the sharp Jackson inequality for the best approximation of f ∈L2,k（Rd） by a subspace Ek2（σ） （SEk2（σ））, which is a subspace of entire functions of exponential type （spherical exponential type） at most σ. Here L2,k（Rd） denotes the space of all d-variate functions f endowed with the L2-norm with the weight vk（x）=Пζ∈R＋}（ζ,x）}2k（ζ）,which is defined by a positive subsystem R＋ of a finite root system R Rd and a function k（ζ）：R→R＋ invariant under the reflection group G（R） generated by R. In the case G（R） = Z2d, we get some exact results. Moreover, the deviation of best approximation by the subspace Ek2（σ） （SE2（σ）） of some class of the smooth functions in the space L2,k（Rd） is obtained.
基金the National Natural Science Foundation of China(No.11371258).
文摘We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator D in the upper half-plane R2+=R×(0,∞),where(Df)(x)=f′0(x)+(λ/x)[f(x)-f(-x)]for givenλ≥0.A C2 function u in R2+is said to beλ-harmonic if(D2x+■2y)u=0.For aλ-harmonic function u in R2+and for a subset E of■R2+=R symmetric about y-axis,we prove that the following three assertions are equivalent:(i)u has a finite non-tangential limit at(x,0)for a.e.x∈E;(ii)u is non-tangentially bounded for a.e.x∈E;(iii)(Su)(x)<∞for a.e.x∈E,where S is a Lusin-type area integral associated with the Dunkl operator D.
