On a condition that a trigonometrical series should have a certain form

Abstract

1. In a recent communication to the Society I have illustrated the fact that the derived series of the Fourier series of functions of bounded variation play a definite part in the theory of Fourier series. Some of the more interesting theorems in that theory can only be stated in all their generality when the coefficients of such derived series take the place of the Fourier constants of a function. I have also recently shown that Lebesgue’s theorem, whether in its original or in its extended form, with regard to the usual convergence of a Fourier series when summed in the Cesàro manner is equally true for the derived series of Fourier series of functions of bounded variation. I have also pointed out that, in considering the effect of all known convergence factors in producing usual convergence, it is immaterial whether the series considered be a Fourier series, or such a derived series. We are thus led to regard the derived series of the Fourier series of functions of bounded variation as a kind of pseudo-Fourier series, possessing properly so-called. In particular we are led to ask ourselves what is the necessary and sufficient condition that a trigonometrical series should have the form in question. One answer is of course immediate. The integrated series must converge to a function of bounded variation. This is merely a statement in slightly different language of the property in question. We require a condition of a simpler formal character, one which does not require us to solve the difficult problem as to whether an assigned trigonometrical series not only converges but also has for sum a function of bounded variation.