On a class of parametric integrals and their application in the theory of fourier series

Abstract

§ 1. The theorem commonly known as Parseval’s Theorem, which, in its latest form, as extended by Fatou, asserts that if f ( x ) and g ( x ) are two functions whose squares are summable, and whose Fourier constants are a n , b n and α n , β n , then the series ½ a 0 α 0 + Ʃ n= 1 ( a n α n + b n β n ) converges absolutely and has for its sum 1/π∫ π -π f ( x ) g ( x ) dx , must he regarded as one of the most important results in the whole of the theory of Fourier series. I have recently, in the 'Proceedings' of this Society and elsewhere, had occasion to illustrate its usefulness, as well as that of certain analogous results to which I have called attention. They may be said, indeed, to have reduced the question of the convergence of Fourier series to the second plane. If we know that a trigonometrical series is a Fourier series, it is in a great variety of cases, embracing even some of the less usual ones, as well as those which ordinarily present themselves, all that we require. It has seemed to me, therefore, worth while to add another to the list of these results. This is the main object of the present paper, in which it is shown that if one of the functions has its (1+ p )th power summable and the other its (1+1/ p )th power summable, where p is any positive quantity, however small, then the above theorem is true with this modification, provided only the series in question is summed in the Cesaro way. In particular, the equality always holds in the ordinary sense whenever the series does not oscillate.