On Fourier series and functions of bounded variation

Abstract

1. In a previous communication to the Society I have pointed out that the succession of constants obtained by multiplying together two successions of Fourier constants in the manner which naturally suggests itself is a succession of Fourier constants, and I have discussed the summability of the function with new constants are associated. We may express the matter in another way by saying that I have shown that the use of the Fourier constants of an even function g(x) as convergence factors in the Fourier series of a function f(x) changes the latter series into a series which is associated with the new series is increased. The use of the Fourier constants of an odd function as convergence factors, on the other hand, has the effect of changing the allied series of the Fourier series of f (x) into a Fourier series, even when the allied series is not itself a Fourier series. It at once suggests itself that the former of the two statements in this form of the result is not the most that can be said. Indeed, the series, whose general term is cos nx , and whose coefficients are accordingly unity, may clearly take the place of the Fourier series of g(x) , although it is not a Fourier series. On the other hand, it is the derived series of the Fourier serious of a function of bounded variation, which is, moreover, odd. We are thus led to ask ourselves whether this is not the trivial case of a general theorem. In the present communication I propose to show, among other things, that the answer to this questions is in affirmative. The following theorems are, in fact, true:—