Matematičeskie zametki, Tome 8 (1970) no. 5, pp. 607-618
Citer cet article
B. V. Pannikov. Convergence of Riemann sums for functions which can be represented by trigonometric series with coefficients forming a monotonic sequence. Matematičeskie zametki, Tome 8 (1970) no. 5, pp. 607-618. http://geodesic.mathdoc.fr/item/MZM_1970_8_5_a7/
@article{MZM_1970_8_5_a7,
author = {B. V. Pannikov},
title = {Convergence of {Riemann} sums for functions which can be represented by trigonometric series with coefficients forming a~monotonic sequence},
journal = {Matemati\v{c}eskie zametki},
pages = {607--618},
year = {1970},
volume = {8},
number = {5},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_5_a7/}
}
TY - JOUR
AU - B. V. Pannikov
TI - Convergence of Riemann sums for functions which can be represented by trigonometric series with coefficients forming a monotonic sequence
JO - Matematičeskie zametki
PY - 1970
SP - 607
EP - 618
VL - 8
IS - 5
UR - http://geodesic.mathdoc.fr/item/MZM_1970_8_5_a7/
LA - ru
ID - MZM_1970_8_5_a7
ER -
%0 Journal Article
%A B. V. Pannikov
%T Convergence of Riemann sums for functions which can be represented by trigonometric series with coefficients forming a monotonic sequence
%J Matematičeskie zametki
%D 1970
%P 607-618
%V 8
%N 5
%U http://geodesic.mathdoc.fr/item/MZM_1970_8_5_a7/
%G ru
%F MZM_1970_8_5_a7
The following theorem is proved. If $$ f(x)=\frac{a_0}2\sum_{k=1}^\infty a_k\cos2\pi kx+b_k\sin2\pi kx $$ where $a_k\downarrow0$ and $b_k\downarrow0$, then $$ \lim_{n\to\infty}\frac1n\sum_{s=0}^{n-1}f\left(x+\frac sn\right)=\frac{a_0}2 $$ on $(0,1)$ in the sense of convergence in measure. If in addition $f(x)\in L^2(0,1)$, then this relation holds for almost all $x$.
