Heat and Poisson semigroups for Fourier-Neumann expansions

Given $\alpha > -1$, consider the second order differential operator in $(0,\infty)$, $$L_\alpha f \equiv (x^2 \frac{d^2}{dx^2} + (2\alpha+3)x \frac{d}{dx} + x^2 + (\alpha+1)^2)(f),$$ which appears in the theory of Bessel functions. The purpose of this paper is to develop the corresponding harmonic analysis taking $L_\alpha$ as the analogue to the classical Laplacian. Namely we study the boundedness properties of the heat and Poisson semigroups. These boundedness properties allow us to obtain some convergence results that can be used to solve the Cauchy problem for the corresponding heat and Poisson equations.Comment: 16 page

Euler's product expansion for the sine: An elementary proof

We provide, by using elementary tools, a new proof of Euler's product expansion for the sine. © The Mathematical Association of America

Weighted inequalities for the Bochner-Riesz means related to the Fourier-Bessel expansions

We prove weighted inequalities for the Bochner-Riesz means for Fourier-Bessel series with more general weights w (x) than previously considered power weights. These estimates are given by using the local Ap theory and Hardy's inequalities with weights. Moreover, we also obtain weighted weak type (1, 1) inequalities. The case when w (x) = xa is sketched and follows as a corollary of the main result. © 2006 Elsevier Inc. All rights reserved

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 L2((-1, 1), |x|2+1 dx). This orthonormal system is a generalization of the classical exponential system defining Fourier series. © 2007 American Mathematical Society

Fourier-Jacobi expansions in Morrey spaces

In this paper we obtain a characterization of the convergence of the partial sum operator related to Fourier-Jacobi expansions in Morrey spaces. © 2014 Elsevier Inc. All rights reserved

Two-weight mixed norm estimates for a generalized spherical mean radon transform acting on radial functions

We investigate a generalized spherical means operator, in other words the generalized spherical mean Radon transform, acting on radial functions. We establish an integral representation of this operator and find precise estimates of the corresponding kernel. As the main result, we prove two-weight mixed norm estimates for the integral operator, with general power weights involved. This leads to weighted Strichartz-Type estimates for solutions to certain Cauchy problems for classical Euler{Poisson{Darboux and wave equations with radial initial data. © 2017 Society for Industrial and Applied Mathematics

Mean and almost everywhere convergence of Fourier-Neumann series

Let J denote the Bessel function of order . The functions x-/2-/2-1/2J++2n+1(x 1/2), n=0,1,2,..., form an orthogonal system in L2((0,),x+dx) when +-1. In this paper we analyze the range of p, , and for which the Fourier series with respect to this system converges in the Lp((0,),xdx)-norm. Also, we describe the space in which the span of the system is dense and we show some of its properties. Finally, we study the almost everywhere convergence of the Fourier series for functions in such spaces. © 1999 Academic Press

A Simple Computation of zeta (2k)

We present a new simple proof of Euler's formulas for zeta(2k), where k = 1, 2, 3,.... The computation is done using only the defining properties of the Bernoulli polynomials and summing a telescoping series. The same method also yields integral formulas for zeta(2k + 1)