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 uncertainty bounds and growth estimates for fractional fourier transforms

Gelfand–Shilov spaces are spaces of entire functions defined in terms of a rate of growth in one direction and a concomitant rate of decay in an orthogonal direction. Gelfand and Shilov proved that the Fourier transform is an isomorphism among certain of these spaces. In this article we consider mapping properties of fractional Fourier transforms on Gelfand–Shilov spaces. Just as the Fourier transform corresponds to rotation through a right angle in the phase plane, fractional Fourier transforms correspond to rotations through intermediate angles. Therefore, the aim of fractional Fourier estimates is to set up a correspondence between growth properties in the complex plane versus decay properties in phase space.     

Wavelet transform associated with linear canonical Hankel transform

The main goal of this paper is to study about the continuous as well as discrete wavelet transform in terms of linear canonical Hankel transform (LCH‐transform) and discuss some of its basic properties. Parseval's relation and reconstruction formula of continuous linear canonical Hankel wavelet transform (CLCH‐wavelet transform) is obtained. Moreover, semidiscrete and discrete LCH‐wavelet transform are also discussed.     

On the Sobolev boundedness results of the product of pseudo-differential operators involving a couple of fractional Hankel transforms

This article is concerned with the study of pseudo-differential operators associated with fractional Hankel transform. The product of two fractional pseudo-differential operators is defined and investigated its basic properties on some function space. It is shown that the pseudo-differential operators and their products are bounded in Sobolev type spaces. Particular cases are discussed.     

Fractional Levy processes on Gel'fand triple and stochastic integration

In this paper, we investigate the long-range dependence of fractional Levy processes on Gel’fand triple and construct stochastic integral with respect to fractional Levy processes for a class of deterministic integrands.     

On a method for solving the inverse scattering problem on the line

A method for solving the inverse scattering problem on the line is proposed. It is based on a Fourier‐Laguerre series representation of the integral transmutation kernel. Substitution of the representation into the Gel'fand‐Levitan‐Marchenko equation leads to a linear algebraic system of equations and consequently to a simple algorithm for recovering the potential.     

Instability in the Gel'fand inverse problem at high energies

We give an instability estimate for the Gel'fand inverse boundary value problem at high energies. Our instability estimate shows an optimality of several important preceding stability results on inverse problems of such a type.     

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.     

Cuspidal Modules as Summands of a Gel'fand–Graev Module

Abstract

Let G = GL _n (q) be the general linear group over a finite field 𝔽 _q with q elements. We call a Gel'fand–Graev module to be the module which affords the Gel'fand–Graev character defined in Definition I.1. It is known that every cuspidal module of G is isomorphic to a (unique) direct summand of a Gel'fand–Graev module. In this article, we investigate a certain endomorphism so that each irreducible cuspidal module is contained in a certain eigenspace corresponding to the cuspidal character. Furthermore, we determine the eigenvalue of that endomorphism by using character theory of finite general linear group.     

Truncations of multilinear Hankel operators

We extend to multilinear Hankel operators the fact that some truncations of bounded Hankel operators are still bounded. We prove and use a continuity property of bilinear Hilbert transforms on products of Lipschitz spaces and Hardy spaces.

     

The transmutation operator method for efficient solution of the inverse Sturm‐Liouville problem on a half‐line

The inverse Sturm‐Liouville problem on a half‐line is considered. With the aid of a Fourier‐Legendre series representation of the transmutation integral kernel and the Gel'fand‐Levitan equation, the numerical solution of the problem is reduced to a system of linear algebraic equations. The potential q is recovered from the first coefficient of the Fourier‐Legendre series. The resulting numerical method is direct and simple. The results of the numerical experiments are presented.     

The Infinite-Dimensional Widths and Optimal Recovery of Generalized Besov Classes

The purpose of the present paper is to consider some weak asymptotic problems concerning the infinite-dimensional Kolmogorov widths, the infinite-dimensional linear widths, the infinite-dimensional Gel'fand widths and optimal recovery in Besov space. It is obvious to be found that the results obtained and methods used in Besov space are easily generalized, and hence in this paper these results in an extension of Besov spaces on R^d are stated and proved. Meantime, the representation theorem and approximation of these spaces by polynomial splines are discussed.     

Uncertainty principles for the continuous Hankel Wavelet transform

The aim of this paper is to prove Heisenberg-type uncertainty principles for the continuous Hankel wavelet transform. We also analyse the concentration of this transform on sets of finite measure. Benedicks-type uncertainty principle is given.     

Continuous and discrete Bessel wavelet transforms

Using the theory of Hankel convolution, continuous and discrete Bessel wavelet transforms are defined. Certain boundedness results and inversion formula for the continuous Bessel wavelet transform are obtained. Important properties of the discrete Bessel wavelet transform are given.     

Simple associative conformal algebras of linear growth

We describe simple finitely generated associative conformal algebras of Gel'fand–Kirillov dimension one.     

Uncertainty principles for the continuous wavelet transform in the Hankel setting

Shapiro’s dispersion and Umbrella theorems are proved for the continuous Hankel wavelet transform. As a side results, we extend local uncertainty principles for set of finite measure to the latter transform.     

Filtrations de Radon et fonctions sphériques pour Sn et son q-analogue GL(n,q)

We define a sequence of generalized Radon transforms, which are intertwining operators for natural representations associated to Gel'fand spaces for the symmetric group S_n. This sequence enables us to decompose in a recursive way these natural representations and to compute explicitly the associate spherical functions. We prove analogous results for a sequence of generalized Radon transforms between natural representations for the general linear group GL(n,q), which are a q-analogue of the preceding ones.     

Weighted generalized functions and hyperbolic Riesz B‐potential

The paper is devoted to the study of the fractional integral operator, which is a negative real power of the singular wave operator generated by Bessel operator using weighted generalized functions. We give conditions for this operator to be bounded in appropriate spaces, obtain formula for the Hankel transform of this operator, and get formula of connection between this operator and natural degree of singular wave operator generated by Bessel operator.     

Diameters of some classes of differentiable periodic functions in the space L

In this paper, diameters in the sense of A. N. Kolmogorov are found for the class of 2π-periodic functions W^(r)H_ω in the space L, that is, d_2n-1(W^(r)H_ω,L), where ω(t) is an upper-convex regular modulus of continuity (r, n=1, 2, ...). An estimate from below is found for diameters in the sense of I. M. Gel'fand, that is, d^2n-!(W^(r)H_ω,L).     

Relating transplantation and multipliers for Dunkl and Hankel transforms

By expressing the Dunkl transform of order α of a function f in terms of the Hankel transforms of orders α and α + 1 of even and odd parts of f, respectively, we show that a considerable part of harmonic analysis of the Dunkl transform on the real line may be reduced to known results for the Hankel transform. In particular, defining the modified Dunkl transform and then considering the Dunkl transplantation operator we transfer known multiplier results for the Hankel transform to the Dunkl transform setting. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)