# On convergence of orthogonal series of Bessel functions

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## Abstract

On convergence of orthogonal series of Bessel functions ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze A. BENEDEK R. PANZONE On convergence of orthogonal series of Bessel functions Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 3e série, tome 27, no 3 (1973), p. 507-525. <http://www.numdam.org/item?id=ASNSP_1973_3_27_3_507_0> © Scuola Normale Superiore, Pisa, 1973, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ ON CONVERGENCE OF ORTHOGONAL SERIES OF BESSEL FUNCTIONS by A. BENEDEK and R. PANZONE ABSTRACT - SUMMARY. Let v be a real number and jtp (x)) a system of solutions of Beaeel’seqnation (ac~’)’+(~ac2013y*/ac)y==0,0~ac, which satisfy a real boundary con- dition at x =1: If it is orthogonal and complete with respect to the measure xdx, then it coincides with the Bessel or Dini system for - 1 v oo or with one of the systems where is the set of zeroes of a certain function associated to the system under con- sideration. Let = (v - 1/2) v 0, 1 + v &#x3E; 0, and 1/2 - fl + ~u 1/p 3/2 2013 ~ 2013 ~ 1 p oo, If Sn (f, Z) denotes the partial sum of the Fourier expansion of f with re- speot to the system (2) and the measure xdx, and ,Sn ( f, x) the partial sum of the expan- sion with respect to the system of Bessel functions satisfying (1) and of order - v, then This is achieved with an estimation of the difference of the Dirichlet kernels. 1. Introduction. Let us consider Bessel’s equation Perv

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