Strong annihilating pairs for the Fourier-Bessel transform

The aim of this paper is to prove two new uncertainty principles for the Fourier-Bessel transform (or Hankel transform). The first of these results is an extension of a result of Amrein, Berthier and Benedicks, it states that a non-zero function f and its Fourier-Bessel transform Fα(f) cannot both have support of finite measure. The second result states that the supports of f and Fα(f) cannot both be (ε,α)-thin, this extending a result of Shubin, Vakilian and Wolff. As a side result we prove that the dilation of a C₀-function are linearly independent. We also extend Faris's local uncertainty principle to the Fourier-Bessel transform.     

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.      

Weyl transforms associated with the Hankel transform in Clifford analysis

In this paper, we define the Hankel–Wigner transform in Clifford analysis and therefore define the corresponding Weyl transform. We present some properties of this kind of Hankel–Wigner transform, and then give the criteria of the boundedness of the Weyl transform and compactness on the L^p space. Copyright © 2005 John Wiley & Sons, Ltd.     

M-harmonic Besovp-spaces and Hankel operators in the Bergman space on the ball in ℂ n

In this paper, we characterize the Hardy class ofM-harmonic functions on the unit ballB in ℂ ⁿ in terms of the Berezin transform. We define and study the Besovp-spaces ofM-harmonic functions. For anM-harmonic symbolf, we give various criteria for the Hankel operatorsH _f andH _f to be bounded, compact or in the Schatten-von-Neumann classS _p . These criteria establish a close relationship among Besovp-spaces, Berezin transform, the invariant Laplacian, and Hankel operators on the unit ballB.     

A transference theorem for the Dunkl transform and its applications

For a family of weight functions invariant under a finite reflection group, we show how weighted Lp multiplier theorems for Dunkl transform on the Euclidean space Rd can be transferred from the corresponding results for h-harmonic expansions on the unit sphere Sd of Rd+1. The result is then applied to establish a Hörmander type multiplier theorem for the Dunkl transform and to show the convergence of the Bochner-Riesz means of the Dunkl transform of order above the critical index in weighted Lp spaces.     

On the Laplace Transform of Radial Functions

Let ?(t) (t ∈ R ⁿ be a radial function. Let f(z) be the Laplace transform of ?(t). Then a theorem due to A. Gonzá Domínguez shows that f(z) can be expressed as a Hankel transform. I prove two representation formulae which express the Laplace transform of radial functions by means of the mth-order derivative of the Hankel transform of order 0 and ? ½.     

Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions

In this paper, we develop two practical methods for the computation of the eigenvalues as well as the eigenfunctions of the finite Hankel transform operator. These different eigenfunctions are called circular prolate spheroidal wave functions (CPSWFs). This work is motivated by the potential applications of the CPSWFs as well as the development of practical methods for computing their values. Also, in this work, we should prove that the CPSWFs form an orthonormal basis of the space of Hankel band-limited functions, an orthogonal basis of L²([0,1]) and an orthonormal system of L²([0,+∞[). Our computation of the CPSWFs and their associated eigenvalues is done by the use of two different methods. The first method is based on a suitable matrix representation of the finite Hankel transform operator. The second method is based on the use of an efficient quadrature method based on a special family of orthogonal polynomials. Also, we give two Maple programs that implement the previous two methods. Finally, we present some numerical results that illustrate the results of this work.     

Hankel transforms of generalized Motzkin numbers

We study Hankel transform of the sequences (u,l,d),t, and the classical Motzkin numbers. Using the method based on orthogonal polynomials, we give closed‐form evaluations of the Hankel transform of the aforementioned sequences, sums of two consecutive, and shifted sequences. We also show that these sequences satisfy some interesting convolutional properties. Finally, we partially consider the Hankel transform evaluation of the sums of two consecutive shifted (u,l,d)‐Motzkin numbers. Copyright © 2017 John Wiley & Sons, Ltd.     

Paley type inequality on the Hardy type space in the Dunkl setting

We investigate the Dunkl transform F_k{\mathcal{F}_k} on Hardy type space in the Dunkl setting and establish a version of Paley type inequality for this transform.     

Multiplier operators and extremal functions related to the dual dunkl-sonine operator

We study the dual Dunkl-Sonine operator ^tS_k,? on ?^d, and give expression of ^tS_k,?, using Dunkl multiplier operators on ?^d. Next, we study the extremal functions f^*_λ, λ >0 related to the Dunkl multiplier operators, and more precisely show that {f^*_λ} λ >0 converges uniformly to ^tS_k,?(f) as λ → 0+. Certain examples based on Dunkl-heat and Dunkl-Poisson kernels are provided to illustrate the results.     

A certain family of fractional wavelet transformations

In the present paper, a fractional wavelet transform of real order α is introduced, and various useful properties and results are derived for it. These include (for example) Perseval's formula and inversion formula for the fractional wavelet transform. Multiresolution analysis and orthonormal fractional wavelets associated with the fractional wavelet transform are studied systematically. Fractional Fourier transforms of the Mexican hat wavelet for different values of the order α are compared with the classical Fourier transform graphically, and various remarkable observations are presented. A comparative study of the various results, which we have presented in this paper, is also represented graphically.     

Convolution operator and maximal function for the Dunkl transform

For a family of weight functionsh _K invariant under a finite reflection group onR ^d, analysis related to the Dunkl transform is carried out for the weightedL ^p spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a maximal function and use it to prove the almost everywhere convergence. ST wishes to thank YX for the warm hospitality during his stay in Eugene. The work of YX was supported in part by the National Science Foundation under Grant DMS-0201669.     

Inversion Formula for the Dunkl-Wigner Transform and Compactness Property for the Dunkl-Weyl Transforms

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 W_σ associated with the Dunkl operators,where cr is a symbol in the Schwartz space S(R~d×R~d).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 W_σ.     

Harmonic Analysis Associated with the One-Dimensional Dunkl Transform

This paper presents a systematic study for harmonic analysis associated with the one-dimensional Dunkl transform, which is based upon the generalized Cauchy–Riemann equations D _x u?? _y v=0,? _y u+D _x v=0, where D _x is the Dunkl operator (D _x f)(x)=f′(x)+(λ/x)(f(x)?f(?x)). Various properties about the λ-subharmonic function, the λ-Poisson integral, the conjugate λ-Poisson integral, and the associated maximal functions are obtained, and the λ-Hilbert transform , a crucial analog to the classical one, is introduced and studied by a stringent method. The theory of the associated Hardy spaces $H_{\lambda}^{p}({\mathbb{R}}^{2}_{+})$ on the half-plane ${\mathbb{R}}^{2}_{+}$ for p≥p ₀=2λ/(2λ+1) with λ>0 extends the results of Muckenhoupt and Stein about the Hankel transform to a general case and contains a number of further results. In particular, the λ-Hilbert transform is shown to be a bounded mapping from $H_{\lambda}^{1}({\mathbb{R}})$ to $L^{1}_{\lambda}({\mathbb{R}})$ ; and associated to the Dunkl transform, an analog of the well-known Hardy inequality is proved for $f\in H^{1}_{\lambda}({\mathbb{R}})$ .     

Markov Processes Related with Dunkl Operators

Dunkl operators are parameterized differential-difference operators on ^Nthat are related to finite reflection groups. They can be regarded as a generalization of partial derivatives and play a major role in the study of Calogero–Moser–Sutherland-type quantum many-body systems. Dunkl operators lead to generalizations of various analytic structures, like the Laplace operator, the Fourier transform, Hermite polynomials, and the heat semigroup. In this paper we investigate some probabilistic aspects of this theory in a systematic way. For this, we introduce a concept of homogeneity of Markov processes on ^Nthat generalizes the classical notion of processes with independent, stationary increments to the Dunkl setting. This includes analogues of Brownian motion and Cauchy processes. The generalizations of Brownian motion have the càdlàg property and form, after symmetrization with respect to the underlying reflection groups, diffusions on the Weyl chambers. A major part of the paper is devoted to the concept of modified moments of probability measures on ^Nin the Dunkl setting. This leads to several results for homogeneous Markov processes (in our extended setting), including martingale characterizations and limit theorems. Furthermore, relations to generalized Hermite polynomials, Appell systems, and Ornstein–Uhlenbeck processes are discussed.     

The Distributional Hankel Transform of Marcel Riesz's Ultrahyperbolic Kernel

In this article we give a sense to the distributional Hankel transform of Marcel Riesz's ultrahyperbolic kernel. First we evaluate Rα(u) in α = ?2k and α = 2k for the cases μ even and ν odd, μ even and ν even, and μ odd and ν odd, μ odd and ν even, where and Finally in Section 4 we obtain the distributional Hankel transform of Marcel Riesz's ultrahyperbolic kernel.     

On the boundedness result of wavelet transform associated with fractional Hankel transform

This article is concerned with the study of the continuity of wavelet transform involving fractional Hankel transform on certain function spaces. The n-dimensional boundedness property of the fractional wavelet transform is also discussed on Sobolev type space. Particular cases are also considered.     

On the norm of the Lp-Dunkl transform

In this work, we establish a Babenko–Beckner-type inequality for the Dunkl transform of a radial function and Dunkl transform associated to a finite reflection group generated by the sign changes. As applications, we establish a new version of Young’s-type inequality and uncertainty relation for Renyi entropy.     

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.     

On Segal�CBargmann analysis for finite Coxeter groups and its heat kernel

We prove identities involving the integral kernels of three versions (two being introduced here) of the Segal?CBargmann transform associated to a finite Coxeter group acting on a finite dimensional, real Euclidean space (the first version essentially having been introduced around the same time by Ben Sa?d and ?rsted and independently by Soltani) and the Dunkl heat kernel, due to R?sler, of the Dunkl Laplacian associated with the same Coxeter group. All but one of our relations are originally due to Hall in the context of standard Segal?CBargmann analysis on Euclidean space. Hall??s results (trivial Dunkl structure and arbitrary finite dimension) as well as our own results in???-deformed quantum mechanics (non-trivial Dunkl structure, dimension one) are particular cases of the results proved here. So we can understand all of these versions of the Segal?CBargmann transform associated to a Coxeter group as Hall type transforms. In particular, we define an analogue of Hall??s Version C generalized Segal?CBargmann transform which is then shown to be Dunkl convolution with the Dunkl heat kernel followed by analytic continuation. In the context of Version C we also introduce a new Segal?CBargmann space and a new transform associated to the Dunkl theory. Also we have what appears to be a new relation in this context between the Segal?CBargmann kernels for Versions A and C.