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 Mat. Zametki: Year: Volume: Issue: Page: Find

 Mat. Zametki, 1970, Volume 8, Issue 5, Pages 607–618 (Mi mz7008)

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
$$\lim_{n\to\infty}\frac1n\sum_{s=0}^{n-1}f(x+\frac sn)=\frac{a_0}2$$
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 $x$.

Full text: PDF file (610 kB)

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

Bibliographic databases:

UDC: 517.5

Citation: B. V. Pannikov, “Convergence of Riemann sums for functions which can be represented by trigonometric series with coefficients forming a monotonic sequence”, Mat. Zametki, 8:5 (1970), 607–618; Math. Notes, 8:5 (1970), 810–816

Citation in format AMSBIB
\Bibitem{Pan70} \by B.~V.~Pannikov \paper Convergence of Riemann sums for functions which can be represented by trigonometric series with coefficients forming a~monotonic sequence \jour Mat. Zametki \yr 1970 \vol 8 \issue 5 \pages 607--618 \mathnet{http://mi.mathnet.ru/mz7008} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=277990} \zmath{https://zbmath.org/?q=an:0216.09401|0205.07304} \transl \jour Math. Notes \yr 1970 \vol 8 \issue 5 \pages 810--816 \crossref{https://doi.org/10.1007/BF01146937}