Mario Pérez

Two notes on convergence and divergence a.e. of Fourier series with respect to some orthogonal systems

We study some problems related to convergence and divergence a.e. for Fourier series in systems {φk}, where {φk} is either a system of orthonormal polynomials with respect to a measure dμ on [−1, 1] or a Bessel system on [0, 1]. We obtain boundedness in weighted L spaces for the maximal operators associated to Fourier-Jacobi and Fourier-Bessel series. On the other hand, we find general results about divergence a.e. of the Fourier series associated to Bessel systems and systems of orthonormal polynomials on [−1, 1]. §0. Introduction. Let dμ be a positive measure on a finite interval [a, b] ⊂ R and

Asymptotic behaviour of orthogonal polynomials relative to measures with mass points

Abstract. General expressions are found for the orthonormal polynomials and the kernels relative to measures on the real line of the form μ+Mδc, in terms of those of the measures dμ and (x−c)dμ. In particular, these relations allow us to obtain that Nevai’s class M(0, 1) is closed for adding a mass point, as well as several bounds for the polynomials and kernels relative to a generalized Jacobi weight with a finite number of mass points.

Weighted Norm Inequalities for Polynomial Expansions Associated to Some Measures with Mass Points

Fourier series in orthogonal polynomials with respect to a measurev on [−1, 1] are studied whenv is a linear combination of a generalized Jacobi weight and finitely many Dirac deltas in [−1, 1]. We prove some weighted norm inequalities for the partial sum operatorsSn, their maximal operatorS*, and the commutator [Mb, Sn], whereMb denotes the operator of pointwise multiplication byb ∈BMO. We also prove some norm inequalites forSn whenv is a sum of a Laguerre weitht onR+ and a positive mass on 0.

Weighted Lp-boundedness of Fourier series with respect to generalized Jacobi weights

Let w be a generalized Jacobi weight on the interval [-1,1] and, for each function f, let Snf denote the n-th partial sum of the Fourier series of f in the orthogonal polynomials associated to w. We prove a result about uniform boundedness of the operators Sn in some weighted Lp spaces. The study of the norms of the kernels Kn related to the operators Sn allows us to obtain a relation between the Fourier series with respect to different generalized Jacobi weights.

Mean convergence of Fourier-Dunkl series

Abstract In the context of the Dunkl transform a complete orthogonal system arises in a very natural way. This paper studies the weighted norm convergence of the Fourier series expansion associated to this system. We establish conditions on the weights, in terms of the A p classes of Muckenhoupt, which ensure the convergence. Necessary conditions are also proved, which for a wide class of weights coincide with the sufficient conditions.

Some Conjectures on Wronskian and Casorati Determinants of Orthogonal Polynomials

In this paper, we conjecture some regularity properties for the zeros of Wronskian and Casorati determinants whose entries are orthogonal polynomials. These determinants are formed by choosing orthogonal polynomials whose degrees run on a finite set of nonnegative integers. The case in which such a set is formed by consecutive integers was studied by Karlin and Szegö.

Heat and Poisson Semigroups for Fourier-Neumann Expansions

AbstractGiven

WEAK BEHAVIOUR OF FOURIER-NEUMANN SERIES

\alpha > -1,

Endpoint weak boundedness of some polynomial expansions

consider the second order differential operator in

Bernoulli-Dunkl and Apostol-Euler-Dunkl polynomials with applications to series involving zeros of Bessel functions

(0,\infty)