an:00497630
Zbl 0796.42003
Zhuk, V. V.
On the question of convergence of a trigonometric Fourier series at a point
EN
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).
00016705
1992
j
42A20 42A50
convergence of Fourier series; Tauberian condition; conjugate series
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.
L.Leindler (Szeged)