Convergence of Riemann sums for functions which can be represented by trigonometric series with coefficients forming a~monotonic sequence

B. V. Pannikov

Matematičeskie zametki, Tome 8 (1970) no. 5, pp. 607-618.

Résumé

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 $x$.