On the Magnitude of Fourier Coefficients Ii

Abstract

For functions on [-ιτ,ττ] 2 which are in various generalized variation classes, estimates of the magnitude of the Fourier coefficients are made. The principal result is best possible in a certain sense.The calculation of the order of magnitude of the Fourier coefficients of a function / of bounded variation is usually accomplished by use of integration by parts and the fact that the variation of / is equal to / \df\ [25, vol.1, p.48]. Unfortunately, this method provides no clue as to how to proceed in higher dimensions and for functions of generalized bounded variation.By extending the notion of integration by parts to the double Riemann-Stieltjes integral, Móricz has used this classical method to effect this calculation for functions of Hardy-Krause bounded variation in two variables [12]. Here we will demonstrate a simple method, used by us [17] and by Silei Wang [19] in the one-variable case, which enables us to determine the order of the Fourier coefficients of functions of bounded variation in any finite number of variables and which extends readily to classes of functions of generalized variation, including the W-classes. Using this method, Fülöp and Móricz [11] have since obtained a result which subsumes our first result. They cite a paper of Wu [24], rediscovering this method in the onevariable case.