Absolute summability of Fourier series and orthogonal series
The successive lecture notes in mathematics deal with absolute Nörlund summability \(| N,p_ n|\) and absolute Riesz summability \(| R,P_ n,1|\) for non-absolute convergent Fourier series and orthogonal series. In this study, starting with the classical results of S. Bernstein, A. Zygmund, R. Salem, O. Szász and S. B. Stechkin, the author presents his own resuls (up to 1981), as well as those obtained by Ch. Watari, S. N. Lal, T. Tsuchikura, T. Kano, G. D. Dikshit, M. Izumi, S. Izumi and other mathematicians. The comprehensive references contain 96 positions, mainly sources. This book consists of 8 chapters the titles of which inform us about the essence, namely: Chapter 1, Absolute convergence of orthogonal series (introduction, comparison theorem, weight absolute convergence, convergence of \(\sum | c_ n|^{\beta})\); Chapter 2, Absolute Nörlund summability almost everywhere of Fourier series (introduction, analogous theorem, proof of theorem, approximation property and absolute Nörlund summability); Chapter 3, Absolute Nörlund summability almost everywhere of orthogonal series (introduction, absolute Nörlund summability of orthogonal series, approximation and absolute summability, structure theorem for trigonometric system, Rademacher trigonometric series, impossibility of improvement of conditions); Chapter 4, Absolute Riesz summability almost everywhere of orthogonal series (introduction, equivalence theorem, sufficient conditions, Rademacher trigonometric series, impossibility of improvement of conditions); Chapter 5, Absolute Nörlund summability factors of Fourier series (introduction, summability factors, some applications, other conditions, some corollaries); Chapter 6, Absolute Nörlund summability factors of conjugate series of Fourier series (introduction, conjugate theorems, some lemmas, proofs of theorems, equivalent relation, some applications); Chapter 7, Local property of absolute Riesz summability of Fourier series (introduction, local property, some applications, non-local property, proofs of theorems); Chapter 8, Local property of absolute Nörlund summability of Fourier series (introduction, local property, proofs of theorems, Izumi theorems, proofs of Izumi theorems);
scientific article; zbMATH DE number 3857698