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 Tr. Mat. Inst. Steklova, 2005, Volume 248, Pages 204–222 (Mi tm132)

This article is cited in 1 scientific paper (total in 1 paper)

The Series $\sum\sum\frac{e^{2\pi imnx}}{mn}$ and a Problem of Chowla

K. I. Oskolkov

University of South Carolina

Abstract: The double trigonometric series $U(x):=\sum_{m=1}^\infty\sum_{n=1}^\infty\frac{e^{2\pi imnx}}{\pi mn}$ and $U(\chi,x):=\sum_{m=1}^\infty\sum_{n=1}^\infty\chi_{m,n}\frac{e^{2\pi imnx}}{\pi mn}$ with the hyperbolic phase and coordinate-wise slow multipliers $\chi_{m,n}$ are studied. Complete descriptions of the $\mathcal K$-convergence (summability) sets of the sine series $\Im U(x)$ and the cosine series $\Re U(x)$ are given. The $\mathcal K$-sum of a double series is defined as the common value of the limits of partial sums over expanding families of kites in $\mathbb N^2$. The latter include convex domains in the usual sense, such as rectangles, as well as nonconvex domains, for example, hyperbolic crosses $\{(m,n):1\le mn\le N\}$.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2005, 248, 197–215

Bibliographic databases:
UDC: 517.518.47
Received in September 2004

Citation: K. I. Oskolkov, “The Series $\sum\sum\frac{e^{2\pi imnx}}{mn}$ and a Problem of Chowla”, Studies on function theory and differential equations, Collected papers. Dedicated to the 100th birthday of academician Sergei Mikhailovich Nikol'skii, Tr. Mat. Inst. Steklova, 248, Nauka, MAIK «Nauka/Inteperiodika», M., 2005, 204–222; Proc. Steklov Inst. Math., 248 (2005), 197–215

Citation in format AMSBIB
\Bibitem{Osk05} \by K.~I.~Oskolkov \paper The Series $\sum\sum\frac{e^{2\pi imnx}}{mn}$ and a~Problem of Chowla \inbook Studies on function theory and differential equations \bookinfo Collected papers. Dedicated to the 100th birthday of academician Sergei Mikhailovich Nikol'skii \serial Tr. Mat. Inst. Steklova \yr 2005 \vol 248 \pages 204--222 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm132} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2165929} \zmath{https://zbmath.org/?q=an:1126.40001} \transl \jour Proc. Steklov Inst. Math. \yr 2005 \vol 248 \pages 197--215 

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This publication is cited in the following articles:
1. Balazard M., Martin B., “On Some Approximate Functional Equations Related to the Gauss Transformation”, Aequ. Math., 93:3 (2019), 563–585
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