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On the question of convergence of a trigonometric Fourier series at a point. (English. Russian original) Zbl 0796.42003
Russ. Acad. Sci., Dokl., Math. 46, No. 2, 349-353 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 326, No. 5, 770-775 (1992).
The author establishes certain new necessary and sufficient conditions for the convergence of Fourier series at a point. His conditions have the character of Tauberian theorems in which the Tauberian condition is written in the form \[ \lim_{n \to \infty} \int^ \delta_ 0 \varphi_ x (t)t^{-1} \sin nt dt=0, \] where \(\varphi_ x(t)\) is less restrictive in relation to the original function \(f\) than the requirement that \[ \lim_{n \to \infty}\int^ \delta_ 0 (f(x+t)+f(x-t) - 2t)t^{-1} \sin nt dt=0. \] The author also studies analogous questions in relation to the conjugate series.
42A20 Convergence and absolute convergence of Fourier and trigonometric series
42A50 Conjugate functions, conjugate series, singular integrals