ON ABSOLUTE CONVERGENCE OF FOURIER SERIES

The necessary and sufficient conditions of the absolute convergence of a trigonometric Fourier series are established for continuous 2π-periodic functions which in [0, 2π] have a finite number of intervals of convexity, and whose nth Fourier coefficients are O(ω(1/n; f)/n), where ω(δ; f) is the continuity modulus of the function f . Let ω be an arbitrary modulus of continuity, i.e., a nondecreasing function continuous on [0, 1], ω(0) = 0 and ω(δ1 + δ2) ≤ ω(δ1) + ω(δ2). As usual, denote by Hω the class of all functions f continuous on [0, 2π] for which ω(δ; f) = sup |x1−x2|≤δ |f(x1)− f(x2)| = O(ω(δ)), 0 ≤ δ ≤ 1 (see, for instance, [5, Ch. 3, pp. 150, 157]). Let M be the class of all continuous 2π-periodic functions f for which there exists a partitioning of the segment [0, 2π] by the points 0 = x1(f) < · · · < xm+1(f) = 2π such that f is convex or concave on each segment [xk(f), xx+1(f)], k = 1, . . .m. The Fourier coefficients of a function f with respect to the trigonometric system will be denoted by an = an(f), bn = bn(f). Problems pertaining to the absolute convergence of Fourier series have been studied quite completely (see, for instance, the monographs of Bari [2, Ch. 9], Zygmund [3, Ch. 6], Kahane [4, Ch. 2], and the survey by Guter and Ulyanov [5, p. 391]). This paper deals with some problems of the absolute convergence of trigonometric Fourier series of a function from the class M . The following facts are well known: 1991 Mathematics Subject Classification. 42A28, 42A16.