Estimating functions by partial sums of their Fourier series

Abstract

For classes of functions with convergent Fourier series. the problem of estimating the rate of convergence of the Fourier series has always been of interest. A classical theorem like that of Dirichlet and Jordan for functions of bounded variation (BV) assures the convergence of the Fourier series but gives no estimate of the rate of convergence. Such an estimate was recently provided by Bojanic [ 11. For certain classes of functions of generalized bounded variation the conclusions of the Dirichlet-Jordan theorem also hold. Waterman 15, 7 1 has shown that the class of functions of harmonic bounded variation (HBV) is, in a certain sense, the largest such class. An estimate of the rate of convergence of the Fourier series has been made for certain classes which lie between BV and HBV [2]. Here we consider this problem in greater generality for /IBV classes in that range to obtain a result which includes the previous estimates and allows us to make an estimate for a particular class which is closer to HBV than the classes previously con sidered. If f is a real valued function on the interval (a. 61 and .I = {A,,/ is a nondecreasing sequence of positive numbers such that x l/i,, diverges. we say that f is of /i-bounded variation (,4BV) if the sums