Weighted Function Spaces and Dunkl Transform

We introduce first weighted function spaces on ${\mathbb{R}^d}$ using the Dunkl convolution that we call Besov-Dunkl spaces. We provide characterizations of these spaces by decomposition of functions. Next we obtain in the real line and in radial case on ${\mathbb{R}^d}$ weighted L ^p -estimates of the Dunkl transform of a function in terms of an integral modulus of continuity which gives a quantitative form of the Riemann-Lebesgue lemma. Finally, we show in both cases that the Dunkl transform of a function is in L ¹ when this function belongs to a suitable Besov-Dunkl space.     

Poisson integrals associated to Dunkl operators for dihedral groups

In this paper we study the boundary behavior of Poisson integrals associated to Dunkl differential-difference operators for dihedral groups and the boundary integral representations for functions on the unit disc of annihilated by the Laplace operator corresponding to these differential-difference operators.

     

Hypoelliptic Jacobi–Dunkl Convolution of Distributions

In this paper we give a necessary and sufficient condition in terms of the Jacobi–Dunkl transform in order that a Jacobi–Dunkl convolution of distributions is hypoelliptic.     

A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform

A Whittaker-Shannon-Kotel'nikov sampling theorem related to the Dunkl transform on the real line is proved. To this end we state, in terms of Bessel functions, an orthonormal system which is complete in . This orthonormal system is a generalization of the classical exponential system defining Fourier series.

     

Hypercyclic and Chaotic Convolution Associated with the Jacobi–Dunkl Operator

In this paper, we study the Jacobi–Dunkl convolution operators on some distribution spaces. We characterize the Jacobi–Dunkl convolution operators as those ones that commute with the Jacobi–Dunkl translations and with the Jacobi–Dunkl operators. Also we prove that the Jacobi–Dunkl convolution operators are hypercyclic and chaotic on the spaces under consideration and we give a universality property for the generalized heat equation associated with them.     

Endoprime modules

Summary We study the harmonic analysis associated with the Dunkl operator on C and analyze the hypercyclicity and chaos of Dunkl convolution operators on the space of entire functions on C.     

Calderon’s Reproducing Formula Related to the Dunkl Operator on the Real Line

 We consider a generalized convolution , , on the real line generated by the Dunkl operator

Weighted Dunkl transform inequalities and application on radial Besov spaces

In Dunkl theory on $\mathbb R ^d$ which generalizes classical Fourier analysis, we prove first weighted inequalities for certain Hardy-type averaging operators. In particular, we deduce for specific choices of the weights the $d$ -dimensional Hardy inequalities whose constants are sharp and independent of $d$ . Second, we use the weight characterization of the Hardy operator to prove weighted Dunkl transform inequalities. As consequence, we obtain Pitt’s inequality which gives an integrability theorem for this transform on radial Besov spaces.     

Perturbations of surjective convolution operators

Let and be (ultra)distributions with compact support which have disjoint singular supports. We assume that the convolution operator is surjective when it acts on a space of functions or (ultra)distribu- tions, and we investigate whether the perturbed convolution operator is surjective. In particular we solve in the negative a question asked by Abramczuk in 1984.

     

Equivalence of <Emphasis Type="Italic">K</Emphasis>-functionals and modulus of smoothness constructed by generalized Dunkl translations

In a Hilbert space L _2,α := L ₂(?, |x|^2α+1 dx), α > ? 1/2, we study the generalized Dunkl translations constructed by the Dunkl differential-difference operator. Using the generalized Dunkl translations, we define generalized modulus of smoothness in the space L _2,α . Based on the Dunkl operator we define Sobolev-type spaces and K-functionals. The main result of the paper is the proof of the equivalence theorem for a K-functional and a modulus of smoothness.     

Uncertainty Principles for the Continuous Dunkl Gabor Transform and the Dunkl Continuous Wavelet Transform

In this paper we consider the Dunkl operators T _j , j = 1, . . . , d, on and the harmonic analysis associated with these operators. We define a continuous Dunkl Gabor transform, involving the Dunkl translation operator, by proceeding as mentioned in [20] by C.Wojciech and G. Gigante. We prove a Plancherel formula, an inversion formula and a weak uncertainty principle for it. Then, we show that the portion of the continuous Dunkl Gabor transform lying outside some set of finite measure cannot be arbitrarily too small. Similarly, using the basic theory for the Dunkl continuous wavelet transform introduced by K. Trimèche in [18], an analogous of this result for the Dunkl continuous wavelet transform is given. Finally, an analogous of Heisenberg’s inequality for a continuous Dunkl Gabor transform (resp. Dunkl continuous wavelet transform) is proved.      

Calderon’s Reproducing Formula Related to the Dunkl Operator on the Real Line

 We consider a generalized convolution , , on the real line generated by the Dunkl operator

Convolution and restriction estimates for measures on curves in

We study convolution and Fourier restriction estimates for some degenerate curves in .

     

On Ginzburg's bivariant Chern classes

The convolution product is an important tool in geometric representation theory. Ginzburg constructed the ``bivariant" Chern class operation from a certain convolution algebra of Lagrangian cycles to the convolution algebra of Borel-Moore homology. In this paper we prove a ``constructible function version" of one of Ginzburg's results; motivated by its proof, we introduce another bivariant algebraic homology theory on smooth morphisms of nonsingular varieties and show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation from the Fulton-MacPherson bivariant theory of constructible functions to this new bivariant algebraic homology theory, modulo a reasonable conjecture. Furthermore, taking a hint from this conjecture, we introduce another bivariant theory of constructible functions, and we show that the Ginzburg bivariant Chern class is the unique Grothendieck transformation from to satisfying the ``normalization condition" and that it becomes the Chern-Schwartz-MacPherson class when restricted to the morphisms to a point.

     

Sequential Fourier-Feynman transform, convolution and first variation

Cameron and Storvick introduced the concept of a sequential Fourier-Feynman transform and established the existence of this transform for functionals in a Banach algebra of bounded functionals on classical Wiener space. In this paper we investigate various relationships between the sequential Fourier-Feynman transform and the convolution product for functionals which need not be bounded or continuous. Also we study the relationships involving this transform and the first variation.

     

Compactly bounded convolutions of measures

In this paper we extend classical results concerning generalized convolution structures on measure spaces. Given a locally compact Hausdorff space , we show that a compactly bounded convolution of point masses that is continuous in the topology of weak convergence with respect to can be extended to a general convolution of measures which is separately continuous in the topology of weak convergence with respect to .

     

Characterization of Function Spaces for the Dunkl Type Operator on the Real Line

In this work we consider the Dunkl operator on the real line, defined by $$ {\cal D}_kf(x):=f'(x)+k\dfrac{f(x)-f(-x)}{x},\,\,k\geq0. $$ We define and study Dunkl–Sobolev spaces \(L^p_{n,k}(\mathbb{R})\) , Dunkl–Sobolev spaces \({\cal L}^p_{\alpha,k}(\mathbb{R})\) of positive fractional order and generalized Dunkl–Lipschitz spaces \(\wedge^k_{\alpha,p,q}(\mathbb{R})\) . We provide characterizations of these spaces and we give some connection between them.     

Convolution Products in Naturally Ordered Semigroups

In this paper new equalities between two different convolution products in cancellative naturally ordered semigroups (but not in groups) are given. We also give several applications in particular cases and      

Perturbations of the Haar wavelet by convolution

In this note we show that the standard convolution regularization of the Haar system generates Riesz bases of smooth functions for , providing in this way an alternative to the approach given by Govil and Zalik [Proc. Amer. Math. Soc. 125 (1997), 3363-3370].

     

Convex integral functionals

We study nonlinear integral functionals determined by normal convex integrands. First we obtain expressions for their convex conjugate, their -subdifferential and their -directional derivative. Then we derive a necessary and sufficient condition for the existence of an approximate solution for the continuous infimal convolution. We also obtain general conditions which guarantee the interchangeability of the conditional expectation and subdifferential operators. Finally we examine the conditional expectation of random sets.