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Convergence and integrability for some classes of trigonometric series

Volume 420 / 2003

Živorad Tomovski Dissertationes Mathematicae 420 (2003), 1-65 MSC: 42A20, 42A32, 26D15, 42A16, 42A50. DOI: 10.4064/dm420-0-1

Abstract

\noindent In this work, the theory of $L^1$-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 $L^1$-convergence of a cosine series $${{a_0}\over{2}}+\sum_{n=1}^\infty a_n\cos nx\leqno{\rm (C)}$$ 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 $L^1$-norm if and only if $a_n\log n=o(1)$ as $n\to\infty$. In 1973, S.~A.~Telyakovski\u\i 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 $L^1$-integrability class for series (C). Also, he proved that if $\{a_n\}_{n=0}^\infty\in S$, then the series (C) converges in $L^1$-norm iff $a_n\log n=o(1)$, $n\to\infty$. The class $S$ is usually called the Sidon–Telyakovski\u\i class. Several authors, Boas, Fomin, \v{C}. Stanojević, Bojanić, and others have extended these classical results by addressing one or both of the following two questions: \vskip4pt {(i)} If $\{a_n\}$ belongs to the class BV of null sequences of bounded variation, is (C) the Fourier series of its sum~$f$? {(ii)} If $\{a_n\}\in BV$, is (C) the Fourier series of some function $f\in L^1$ and is it true that $\Vert S_n-f\Vert=o(1)$ as $n\to\infty$ iff $a_n\log n=o(1)$, $n\to\infty$? \vskip4pt Here, $S_n$ denotes the $n$th partial sum of the series (C), and $\Vert\,\cdot\,\Vert$ is the $L^1$-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 $L^1$-convergence of the $r$th derivative of Fourier series, i.e. we define some new integrability classes of the $r$th derivative of Fourier series. Some necessary and sufficient conditions for $L^1$-convergence of the $r$th derivative of Fourier series are obtained. We define some new $L^p$ $(0

Authors

  • Živorad TomovskiDepartment of Mathematics
    Faculty of Mathematical and Natural Sciences
    St. Kiril and Metodij University
    Skopje, Macedonia
    e-mail

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