Order of magnitude of multiple fourier coefficients of functions of bounded variation

Abstract

Let f : R → C be a periodic function with period 2π in each variable and of bounded variation in the sense of Vitali over [0, 2π] with the total variation V ( f ; [0, 2π] ) . In case f is Lebesgue integrable over the n-dimensional torus [0, 2π), denote by f(k1, . . . , kn) the multiple Fourier coefficients of f , where (k1, . . . , kn) ∈ Z. We present a straightforward proof of the estimate ∣∣ f(k1, . . . , kn)∣∣ 5 V (f ; [0, 2π]n) (2π) n ∏ j=1 kj , provided kj 6= 0, j = 1, . . . , n. Both the order of magnitude and the constant in this estimate are exact. 1. Functions of bounded variation Let f = f(x1, . . . , xn) be a real or complex-valued function defined on a rectangle Q with sides parallel to the coordinate axes: Q := { (x1, . . . , xn) ∈ R : aj 5 xj 5 bj ; j = 1, . . . , n } , where −∞ < aj < bj < +∞ for each j. By a (finite) partition P of Q we mean P := P1 × . . .× Pn, where Pj : aj = xj < xj < · · · < x sj j = bj , sj = 1; j = 1, . . . , n. ∗This research was supported by the Hungarian National Foundation for Scientific Research under Grant T 046192.