On the determination of the jump of a function by its Fourier series

Abstract

We shall suppose that I=[0, 2rc] and that f is 2~-periodic. The relationship of this notion of bounded variation to another interesting generalization given by Z. A. ~anturija's modulus of variation [4], was discussed by the author in [1] and [2]. The existing inclusion relations yield that certain theorems on the absolute convergence of Fourier series of functions of Waterman classes {n~}BV, 0 < ~ < 1, are equivalent to the corresponding statements for ~anturija's V[n~], 0 < y < 1, and further to the older ones for Wiener classes of bounded p-variation, [2]. The case c~= 1; i.e. 2 ,=n, when we also speak of harmonic bounded variation (ttBV) is, however, a distinguished one. It appears that some important classical theorems, valid for the Jordan class BV, find in HBV their natural setting. D. Waterman has proved in [9] that the Fourier series of functions of this class converge everywhere and converge uniformly on closed intervals of continuity. (See also ~anturija [5]. Note that his class V[v(n)] with Zv(k)/k~< co is contained in HBV.) Further, the classes L and HBV are complementary in the sense that for a product of fCL and gEHBV the Parseval formula holds (Waterman, [11]). Both of these results are best possible in that they are not valid for any strictly larger class ABV. In this note we shall show this is also the case with the equation