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 Izv. Akad. Nauk SSSR Ser. Mat., 1980, Volume 44, Issue 6, Pages 1255–1278 (Mi izv1961)

On the rate of convergence of integrals of Gauss–Weierstrass type for functions of several variables

B. I. Golubov

Abstract: A one-parameter class of summability methods for multiple Fourier series and Fourier integrals is considered. This class includes the Abel–Poisson method and the Gauss–Weierstrass method. These methods are used to investigate the rate of summability of Fourier series and integrals of differentiable functions. As corollaries, criteria are obtained for harmonicity and polyharmonicity of functions in given domains of a multidimensional Euclidean space. For example, a criterion is obtained for harmonicity and polyharmonicity of a polynomial in $N$ variables. Moreover, the rate of convergence in the $L_p$-metric is studied for singular integrals of the class under discussion for functions in the Nikol'skii class $H_p^\alpha$ ($\alpha>0$, $1\leqslant p\leqslant\infty$).
Bibliography: 14 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1981, 17:3, 455–475

Bibliographic databases:

UDC: 517.5
MSC: Primary 41A25, 42A24, 42B10, 42B20; Secondary 46E35, 47F05

Citation: B. I. Golubov, “On the rate of convergence of integrals of Gauss–Weierstrass type for functions of several variables”, Izv. Akad. Nauk SSSR Ser. Mat., 44:6 (1980), 1255–1278; Math. USSR-Izv., 17:3 (1981), 455–475

Citation in format AMSBIB
\Bibitem{Gol80} \by B.~I.~Golubov \paper On the rate of convergence of integrals of Gauss--Weierstrass type for functions of several variables \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1980 \vol 44 \issue 6 \pages 1255--1278 \mathnet{http://mi.mathnet.ru/izv1961} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=603577} \zmath{https://zbmath.org/?q=an:0506.40002|0479.40003} \transl \jour Math. USSR-Izv. \yr 1981 \vol 17 \issue 3 \pages 455--475 \crossref{https://doi.org/10.1070/IM1981v017n03ABEH001368} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1981NK82000002} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. David J. Marcus, “Non-stable laws with all projections stable”, Z Wahrscheinlichkeitstheorie verw Gebiete, 64:2 (1983), 139
2. Sinem Sezer, Ilham A. Aliev, “On the Gauss-Weierstrass summability of multiple trigonometric series at µ-smoothness points”, Acta. Math. Sin.-English Ser, 27:4 (2011), 741
3. Melih Eryigit, “On degree of approximation of the Gauss-Weierstrass means for smooth Lp(Rn) functions”, J Inequal Appl, 2013:1 (2013), 428
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