This article is cited in 1 paper

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

B. V. Pannikov

V. A. Steklov Mathematical Institute, USSR Academy of Sciences

Abstract: 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 $x$.

UDC: 517.5

Received: 12.12.1969

 English version: Mathematical Notes, 1970, 8:5, 810–816

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