On the summability of the developments in Bessel’s functions

Abstract

at all points at which the function /(Z) is continuous or has a finite jump, if f(Z) is finite and integrable or even if/(Z) becomes infinite at a finite number of points, provided it remains integrable. He llas further shown in the article referred to that the development is uniformly sumrnable throughout any elosed -interYal which does not include a point of discontinuity of the function. IIe has used these properties in connection with a general theorem about convergence factors to establish the faot that the formal results obtained in the discussion of many problems in Mathematical Physics which involve the development of an arbitrary function in a Fourier's series really furnish a solution of the problem, even when the development of the function is a divergent selies. FEJ:ER'S idea of investigating the nature of the divergence of a development fof an arbitrary function in terms of normal functions and then applving general theorems about convergence factors to determine the behavier of the series when such factors are introduced, can be applied to many other developments that ocour in Mathematical Physics. The present paper is devoted to a study, from this point of view, of the developments in terlns of Bessel functiolls. Although the formal work of obtaining the development of an arbitrary function in terms of Bessel functiotls goes back to FOURIER, the rigorous discussion of the conditions under whith the developmellt is convergent has not made equal progress with that discussion in the case of the ordinary Fourier's series. -This is doubtless due to the less elementary character of the functions involved