On the Convergence in Mean of Trigonometric Fourier Series

A. S. Belov

Ivanovo State University

Abstract: We prove the sharpness of Zygmund's theorem, which asserts that if a $2\pi$-periodic function $f$ belongs to $L\ln^+ L$, then its Fourier series is convergent in mean.

Keywords: trigonometric Fourier series, $2\pi$-periodic function, convergence in mean, Zygmund's theorem, Abel transformation, Dirichlet kernel.

UDC: 517.5

Received: 14.01.2009

DOI: 10.4213/mzm8368

 English version: Mathematical Notes, 2010, 87:4, 466–474

Bibliographic databases:

• 
• 
• 
•