On Gibbs’s phenomenon for the developments in Bessel’s functions

Abstract

Introduction. Several investigations have recently been publishedt concerning Gibbs's phenomenon in the case of the developments in Bessel's functions. As we should expect, the behavior of the developments in the neighborhood of the origin presented special difficulty, and the investigators limited themselves to very special types of functions$ in dealing with this particular situation. It is the purpose of the present note to point out that the asymptotic formulas? for the coefficients of the developments in Bessel's functions that have been obtained in two of my previous papers on this subject furnish a simple method of attacking the point in question. From these formulas the nature of Gibbs's phenomenon at the origin can be readily derived for general classes of functions. The same formulas also serve to establish the facts with regard to Gibbs's phenomenon at interior points of the interval (0, 1) and at the end point 1. 1. The cause of Gibbs's phenomenon at the origin. I will illustrate the application of these formulas to the discussion of Gibbs's phenomenon by dealing with the point x = 0 and the class of functions considered in the earlier paper. This class includes functions which in the interval (0, 1) have a second derivative that is finite and integrable, except perhaps for a finite number of points at which the function itself or its first derivative has a finite jump. For such functions we have? for the coefficient of the general term of the development in Bessel's functions the following asymptotic