On the multiplication of successions of Fourier constants

Abstract

1. In a short note to appear in the 'Comptes Rendus’ I have shown, by means of reasoning of a somewhat delicate nature, that, if we multiply together corresponding constants of two successions of Fourier constants, the succession of constants so obtained is the succession of the Fourier constants of a function whose summability depends on those of the two functions to which the two given successions belong in a certain definite way. If (1 + p ) and (1 + q ) represent the indices of the summability of the given functions, that of the new function so obtained is denoted by (1 + p ) (1 + q )/(1- pq ). I have there shown how this theorem may be applied to obtain a notable extension of Parseval’s theorem, and I have briefly indicated that we are thus able to give to that generalisation a still more complete form, namely, that the series Ʃ n=1 ( a n 2k + b n 2k ), where k is any positive integer, converges if the 2 k /(2 k - 1)th power of the function of which a n and b n are the Fourier constants is summable. In the present communication I propose to give the necessary additional reasoning by which this result is obtained, and to deduce other important consequences. For this purpose we require the following generalisation of the first result noted above :—If the denominator of the expression 1 + P = r=m II r=1 (1 + p r ) / [1 - r=m Ʃ r=2 (r - 1) T r ], where T r denotes the sum of the products of the positive quantities p 1 , p 2 ....., p m taken r at a time, is positive, then the succession of constants obtained by multiplying m successions of Fourier constants together is associated with a function whose index of summability is 1 + P, when those of the separate functions are 1 + p r , for all values of r from 1 to m .