On integrability and L1-convergence differentiated trigonometric series

New integrability and L-1-convergence classes for the r-times differentiated trigonometric series Sigma(kis an element ofZ)(ik)(r) c(k)e(ikx) are derived using known results on integrability and L-1-convergence of trigonometric series. These classes, Theorems 1 and 2, subsume all known integrability and L-1-convergence classes for differentiated series. In particular, the extensions of the Fomin type theorems to r-times differentiated series, due to S.S. Bhatia and B. Ram (Proc. Amer. Math. Soc. 124 (1996) 1821-1829) and S.Y. Sheng (Proc. Amer. Math. Soc. 110 (1990) 895-904), are deduced from our Theorem 2 and its corollary Theorem 4. Another extension of the known results for differentiated series is given in Theorem 6. This result is proved using Fomin's theorem on cosine and sine series (Mat. Zametki 23 (1978) 213-222), and as a corollary of Theorem 2. The first proof yields another interesting conclusion, namely that the Fomin type theorems for differentiated series can be deduced as consequences of the original Fornin's results. (C) 2003 Elsevier Inc. All rights reserved.