On 𝐿¹ convergence of certain cosine sums

Abstract

Rees and Stanojevic introduced a new class of modified cosine sums {gn(x) = 2 '=O Aa(k) + 2k=1 2jk Aa(j)cos kx) and found a necessary and sufficient condition for integrability of these modified cosine sums. Here we show that to every classical cosine series f with coefficients of bounded variation, a Rees-Stanojevic cosine sum gn can be associated such that gn converges to f pointwise, and a necessary and sufficient condition for L1 convergence of gn to f is given. As a corollary to that result we have a generalization of the classical result of this kind. Examples are given using the well-known integrability conditions. Theorem A gives a necessary and sufficient condition for a sine series with coefficients of bounded variation and converging to zero to be the Fourier series of its sum, or equivalently, for its sum to be integrable. Theorem B shows that if such a series is a Fourier series then its convergence is good, that is, convergence in the L1 metric. THEOREM A [1]. Let f(x) = 2 ??I b(n)sin nx where l\b(n) > 0 [?Ab(n) = b(n) b(n + 1)] and lim b(n) = 0. Then f E L1 [0, T] or, equivalently, b(n) sin nx is the Fourier series off if and only if n A Ib(n)I log n < oo. THEOREM B [1]. Let f(x) be as in Theorem A. If f E L'[O,] then k= I b(k) sin kx converges to f in the L1 metric. There is no known analogue of Theorem A for the cosine series. Theorems C and D only give sufficient conditions for the cosine series to be the Fourier series of its sum. In what follows we will denote by C the cosine series 00 2a(O) + 2 a(n)cos nx n=l where limn a(n) = 0 and n? Izla(n)l < 00. Partial sums of C will be denoted by Sn (x), and f (x) = limnOo Sn (x) THEOREM C [1]. If n '1 zIla(n) Ilog n < x, thenf E L1 [O, ] or, equivalently, C is the Fourier series off. THEOREM D [1]. If 2 ??1 I \2a(n)I(n + 1) < oo, then f e L1 [0, ] or, equivalently, C is the Fourier series of f. Received by the editors October 21, 1974 and, in revised form, January 9, 1975. AMS (MOS) subject classifications (1970). Primary 42A20, 42A32.