Z. Tomovski
Convergence and Integrability on some Classes of Trigonometric Series
(86 pages in total)
Abstract
In this work, the theory of L^{1}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 wellknown theorems are given. A classical result concerning the
integrability and L^{1} convergence of a cosine series
with convex coefficients is the wellknown 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 L^{1}norm if and only if a_{n} log n = o(1), as n ® ¥. In 1973, S. A. Telyakovskii [44] extended the old classical result of Kolmogorov. Namely, Telyakovskii introduced a Sidontype 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 L^{1}integrability class for series (C). Also, Telyakovskii in his paper showed that such cosine series (C) converge in L^{1}norm iff a_{n} log n = o(1), n ® ¥. The class S is usually called as SidonTelyakovskii 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 {a_{n}} belongs to the class of null sequences of bounded variation BV, then (C) is the Fourier series of its sum function f. (ii) If {a_{n}}Î BV, then (C) is the Fourier series of some function f Î L^{1} and  S_{n } f  = o(1), as n ® iff a_{n} log n = o(1), n ®. Here, S_{n} denotes the nth partial sum of the series (C), and  ·  is the L^{1}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 L^{1}convergence of the rth derivate of Fourier series, i.e. we defined some new integrability classes of the rth derivate of the Fourier series. Some necessary and sufficient conditions for L^{1}convergence of the rth derivate of Fourier series are obtained. Also it is shown a version of Uljanov's theorem [59] and extend it to the rth derivate of trigonometric series. Note: For more information on this thesis,
contact Z. Tomovski.
