PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 66, Number 2, October 1977 CONVERGENCE OF CERTAIN COSINE SUMS IN THE METRIC SPACE L BABU RAM Abstract. We consider here the Ll convergence of Rees-Stanojevic cosine sums to a cosine trigonometric series belonging to the class S defined by Sidon and deduce as corollaries some previously known results from our result. 1. Introduction. Sidon [6] introduced the following class of cosine trigonometric series: Let a0 °° (1.1) -y + 2 ak cos k* L k = X be a cosine series satisfying ak = o(\), k -» oo. If there exists a sequence {Ak} such that (1.2) Ak{0, k-^co, 00 (1.3) 2 Ak < oo, k = 0 (1.4) \àak\<Ak, VA:, we say that (1.1) belongs to the class S. Let the partial sums of (1.1) be denoted by Sn(x) and/(x) = lim„^0O Sn(x). Recently, Garrett and Stanojevic [3] proved that the partial Rees-Stanojevic sums [5] ■ n n n (1.5) g„(x) = - 2 Aß* + 2 2 Aa,- cos kx z Ar = 0 *=1 j = k converge in the L1 metric to (1.1) if and only if given e > 0, there is a <5(e) > 0 such that (1.6) fl S àakDk(x) k = n + X dx< e for all n > 0. It has been shown in the same paper that the classical Young-Kolmogorov-Stanojevic sufficient conditions for integrability of (1.1) imply (1.6). Received by the editors October 7, 1976. AMS (MOS) subject classifications (1970). Primary42A20, 42A32. Key words and phrases. L' convergence of cosine sums, the class 5, quasi-convex sequence. © American Mathematical Society 1977 258 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use