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Convergence and integrability for some classes of trigonometric series

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Tomovski Ž.

Instytut Matematyczny Polskiej Akademii Nauk

In this work, the theory of L¹-convergence for some classes of trigonometric series is elaborated. The work contains four chapters in which some new results are obtained. Also, new proofs of some well known theorems are given. A classical result concerning the integrability and L¹-convergence of a cosine series $a₀/2 + ∑_{n=1}^{∞} aₙ cos nx$ (C) with convex coefficients is the well known theorem of Young. Later, Kolmogorov extended Young's result for series (C) with quasi-convex coefficients and also showed that such cosine series converge in L¹-norm if and only if aₙlog n = o(1) as n → ∞. In 1973, S. A. Telyakovskiĭ extended the old classical result of Kolmogorov. Namely, he redefined the class of numerical sequences introduced by S. Sidon. He denoted this class by S and proved, first, that the Sidon class is equivalent to S, and second, that S is an L¹-integrability class for series (C). Also, he proved that if ${aₙ}_{n=0}^{∞} ∈ S$, then the series (C) converges in L¹-norm iff aₙlog n = o(1), n → ∞. The class S is usually called the Sidon-Telyakovskiĭclass. Several authors, Boas, Fomin, Č. Stanojević, Bojanić, and others have extended these classical results by addressing one or both of the following two questions: (i) If {aₙ} belongs to the class BV of null sequences of bounded variation, is (C) the Fourier series of its sum f? (ii) If {aₙ} ∈ BV, is (C) the Fourier series of some function f ∈ L¹ and is it true that |Sₙ-f| = o(1) as n → ∞ iff aₙlog n = o(1), n → ∞ ? Here, Sₙ denotes the nth partial sum of the series (C), and |·| is the L¹-norm. Fomin, Stanojević, Tanović-Miller and others have given a positive answer to question (ii) for many classes of sequences. We consider the problem of L¹-convergence of the rth derivative of Fourier series, i.e. we define some new integrability classes of the rth derivative of Fourier series. Some necessary and sufficient conditions for L¹-convergence of the rth derivative of Fourier series are obtained. We define some new $L^{p}$ (0 < p < 1) convergence classes of numerical sequences, and extend a theorem of Ul'yanov's theorem by considering the rth derivative of complex trigonometric series.

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1 Tomovski Ž.

1 2003

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Convergence and integrability for some classes of trigonometric series