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On the convergence in mean of trigonometric Fourier series
Mathematical Notes, 2010We prove the sharpness of Zygmund’s theorem, which asserts that if a 2π-periodic function f belongs to L ln+ L, then its Fourier series is convergent in mean.
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ON $ (H,k)$-SUMMABILITY OF MULTIPLE TRIGONOMETRIC FOURIER SERIES
Mathematics of the USSR-Izvestiya, 1977A theorem is proved from which, in particular, it follows that if on , then the multiple trigonometric Fourier series of and all conjugate series are -summable almost everywhere on for every . In the case where this result was obtained by Marcinkiewicz (Collected papers, PWN, Warsaw, 1964).
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On divergence of trigonometric Fourier series everywhere
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 1999The main result of this article is the Theorem: Let an increasing convex function \(\varphi: \mathbb{R}_+\to \mathbb{R}_+= [0,\infty)\) and a sequence \(\{\psi(m)\}\) satisfy the conditions \(\psi(m)= 1\), \(m= 1,2,\dots\), and \(\varphi(m)\psi(m)= o\left({m\sqrt{\log m}\over \sqrt{\log\log m}}\right)(m\to\infty)\).
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On everywhere divergence of trigonometric Fourier series
Sbornik: Mathematics, 2000A general theorem is proved, whose strength is illustrated by the following two corollaries. Corollary 1. If \(\varphi:[0,\infty)\to [0,\infty)\) is a nondecreasing function such that \[ \varphi(u)= o(u\sqrt{\ln u}/\sqrt{\ln\ln\ln u})\quad\text{as}\quad u\to\infty, \] then there exists a function \(f\in L(\mathbb{T})\) such that \[ \int_{\mathbb{T ...
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On Absolute Summability of Factored Infinite Series and Trigonometric Fourier Series
Results in Mathematics, 2018zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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On Trigonometric Fourier Series and Wiener Algebras
Ukrainian Mathematical BulletinThe paper presents new connections between Wiener Banach algebras of absolutely convergent Fourier integrals of complex-valued Borel measures and various issues in the theory of Fourier series and integrals, as outlined in the classical monographs by Zygmund [1], Bary [2], and Stein-Weiss [3]. This two-way connection allows, in particular, deriving new
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The Expression of Trigonometrical Series in Fourier Form
Journal of the London Mathematical Society, 1936openaire +2 more sources