Convergence and Integrability on some Classes of Trigonometric Series
(86 pages in total)
Z. Tomovski
Convergence and Integrability on some Classes of Trigonometric Series
(86 pages in total)
Abstract
In this work, the theory of L1-convergence
on some classes of trigonometric series is elaborated. It 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
with convex coefficients is the well-known theorem of Young [61]. Later, Kolmogorov [21] extends Young's result for series (C) with quasiconvex coefficients and also showed that such cosine series converge in L1-norm if and only if an log n = o(1), as n ® ¥. In 1973, S. A. Telyakovskii [44] extended the old classical result of Kolmogorov. Namely, Telyakovskii introduced a Sidon-type condition [35] described by class S in his paper. Firstly, he proved that the Sidon's class is equivalent to the class S, and second that S is a L1-integrability class for series (C). Also, Telyakovskii in his paper showed that such cosine series (C) converge in L1-norm iff an log n = o(1), n ® ¥. The class S is usually called as Sidon-Telyakovskii class. Several authors, Boas, Fomin, C. Stanojevic, Bojanic, and others have extended these classical results by answering one or both of the following two questions: (i) If {an} belongs to the class of null sequences of bounded variation BV, then (C) is the Fourier series of its sum function f. (ii) If {an}Î BV, then (C) is the Fourier series of some function f Î L1 and || Sn - f || = o(1), as n ® iff an log n = o(1), n ®. Here, Sn denotes the n-th partial sum of the series (C), and || · || is the L1-norm. Namely, these authors, for any classes of sequences have given a positive answer of the question (ii). Then we have considered the problem of L1-convergence of the r-th derivate of Fourier series, i.e. we defined some new integrability classes of the r-th derivate of the Fourier series. Some necessary and sufficient conditions for L1-convergence of the r-th derivate of Fourier series are obtained. Also it is shown a version of Uljanov's theorem [59] and extend it to the r-th derivate of trigonometric series. Note: For more information on this thesis,
contact Z. Tomovski.
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