Jakob Dunkt:querkraft事务所建筑师、CEO.Architect and CEO of querkraft.胡越(HuYue):全国工程勘察设计大师:北京建筑大学教授;北京市建筑设计研究院有限公司总建筑师:《建筑创作》杂志主编。National Engineering Survey and Design...Jakob Dunkt:querkraft事务所建筑师、CEO.Architect and CEO of querkraft.胡越(HuYue):全国工程勘察设计大师:北京建筑大学教授;北京市建筑设计研究院有限公司总建筑师:《建筑创作》杂志主编。National Engineering Survey and Design Master Professor at Beijing University of Civil Engineering and Architecturei Chief Architect of Beijing Institute of Architectural Design Archicreation Journal Chief Editor.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.
We define and study the Fourier-Wigner transform associated with the Dunkl operators, and we prove for this transform an inversion formula. Next, we introduce and study the Weyl transforms Wσ associated with the Dunk...We define and study the Fourier-Wigner transform associated with the Dunkl operators, and we prove for this transform an inversion formula. Next, we introduce and study the Weyl transforms Wσ associated with the Dunkl operators, where a is a symbol in the Schwartz space S（Rd × Rd）. An integral relation between the precedent Weyl and Wigner transforms is given. At last, we give criteria in terms of σ for boundedness and compactness of the transform Wσ.展开更多
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
We consider the harmonic analysis associated with the Dunkl operators on Rd. We study the Dunkl mean-periodic functions on the space ε(Rd) (the space of C∞-functions). We characterize also the continuous linear mapp...We consider the harmonic analysis associated with the Dunkl operators on Rd. We study the Dunkl mean-periodic functions on the space ε(Rd) (the space of C∞-functions). We characterize also the continuous linear mappings from ε(Rd) into itself which commute with the Dunkl operators.展开更多
In this work, we introduce a class of Hilbert spaces of entire functions on the disk , 0<q<1 , with reproducing kernel given by the q-Dunkl kernel . The definition and properties of the space extend naturally th...In this work, we introduce a class of Hilbert spaces of entire functions on the disk , 0<q<1 , with reproducing kernel given by the q-Dunkl kernel . The definition and properties of the space extend naturally those of the well-known classical Fock space. Next, we study the multiplication operator Q by z and the q-Dunkl operator on the Fock space;and we prove that these operators are adjoint-operators and continuous from this space into itself.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
Let Ω be a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (△h)nf =0 for some i...Let Ω be a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (△h)nf =0 for some integer n. Here △h=∑j=1N Dj2 is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G, where kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form f(x)=f0(x)+|x|2f1(x)+…+|x|2(n-1)fn-1(x),(?)x∈Ω, where fj are Dunkl harmonic functions, i.e. △hfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.展开更多
In the framework of superspace in Clifford analysis for the Dunkl version,the Fischer decomposition is established for solutions of the Dunkl super Dirac operators.The result is general without restrictions on multipl...In the framework of superspace in Clifford analysis for the Dunkl version,the Fischer decomposition is established for solutions of the Dunkl super Dirac operators.The result is general without restrictions on multiplicity functions or on super dimensions.The Fischer decomposition provides a module for the Howe dual pair G×osp(1|2)on the space of spinor valued polynomials with G the Coxeter group,while the generators of the Lie superspace reveal the naturality of the Fischer decomposition.展开更多
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
In this paper,the authors obtain the Dunkl analogy of classical L^(p)Hardy inequality for p>N+2γwith sharp constant((p-N-2γ)/p)^(p),where 2γis the degree of weight function associated with Dunkl operators,and L ...In this paper,the authors obtain the Dunkl analogy of classical L^(p)Hardy inequality for p>N+2γwith sharp constant((p-N-2γ)/p)^(p),where 2γis the degree of weight function associated with Dunkl operators,and L pHardy inequalities with distant function in some G-invariant domains.Moreover they prove two Hardy-Rellich type inequalities for Dunkl operators.展开更多
文摘Jakob Dunkt:querkraft事务所建筑师、CEO.Architect and CEO of querkraft.胡越(HuYue):全国工程勘察设计大师:北京建筑大学教授;北京市建筑设计研究院有限公司总建筑师:《建筑创作》杂志主编。National Engineering Survey and Design Master Professor at Beijing University of Civil Engineering and Architecturei Chief Architect of Beijing Institute of Architectural Design Archicreation Journal Chief Editor.
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 the DGRST Research Project LR11ES11 and CMCU Program 10G/1503
文摘We define and study the Fourier-Wigner transform associated with the Dunkl operators, and we prove for this transform an inversion formula. Next, we introduce and study the Weyl transforms Wσ associated with the Dunkl operators, where a is a symbol in the Schwartz space S（Rd × Rd）. An integral relation between the precedent Weyl and Wigner transforms is given. At last, we give criteria in terms of σ for boundedness and compactness of the transform Wσ.
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We consider the harmonic analysis associated with the Dunkl operators on Rd. We study the Dunkl mean-periodic functions on the space ε(Rd) (the space of C∞-functions). We characterize also the continuous linear mappings from ε(Rd) into itself which commute with the Dunkl operators.
文摘In this work, we introduce a class of Hilbert spaces of entire functions on the disk , 0<q<1 , with reproducing kernel given by the q-Dunkl kernel . The definition and properties of the space extend naturally those of the well-known classical Fock space. Next, we study the multiplication operator Q by z and the q-Dunkl operator on the Fock space;and we prove that these operators are adjoint-operators and continuous from this space into itself.
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.
基金supported by the National Natural Science Foundation of China(Grant No.10471134).
文摘Let Ω be a G-invariant convex domain in RN including 0, where G is a Coxeter group associated with reduced root system R. We consider functions f defined in Ω which are Dunkl polyharmonic, i.e. (△h)nf =0 for some integer n. Here △h=∑j=1N Dj2 is the Dunkl Laplacian, and Dj is the Dunkl operator attached to the Coxeter group G, where kv is a multiplicity function on R and σv is the reflection with respect to the root v. We prove that any Dunkl polyharmonic function f has a decomposition of the form f(x)=f0(x)+|x|2f1(x)+…+|x|2(n-1)fn-1(x),(?)x∈Ω, where fj are Dunkl harmonic functions, i.e. △hfj = 0. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition.
基金supported by the Unidade de Investigao"Matemtica e Aplicaoes"of University of AveiroNational Natural Science Foundation of China(Grant No.10771201)support by Ghent University and expresses his sincere gratitude to Prof.F.Brackx,H.De Schepper,and F.Sommen for the kind hospitality during the visit
文摘In the framework of superspace in Clifford analysis for the Dunkl version,the Fischer decomposition is established for solutions of the Dunkl super Dirac operators.The result is general without restrictions on multiplicity functions or on super dimensions.The Fischer decomposition provides a module for the Howe dual pair G×osp(1|2)on the space of spinor valued polynomials with G the Coxeter group,while the generators of the Lie superspace reveal the naturality of the Fischer decomposition.
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supported by the National Natural Science Foundation of China(Nos.11771395,12071431)
文摘In this paper,the authors obtain the Dunkl analogy of classical L^(p)Hardy inequality for p>N+2γwith sharp constant((p-N-2γ)/p)^(p),where 2γis the degree of weight function associated with Dunkl operators,and L pHardy inequalities with distant function in some G-invariant domains.Moreover they prove two Hardy-Rellich type inequalities for Dunkl operators.
