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 Zh. Vychisl. Mat. Mat. Fiz., 2018, Volume 58, Number 10, Pages 1616–1626 (Mi zvmmf10789)

Fourier method for solving two-sided convolution equations on finite noncommutative groups

V. M. Deundyak, D. A. Leonov

Institute of Mathematics, Mechanics, and Computer Science, Southern Federal University, Rostov-on-Don, Russia

Abstract: The Fourier method on commutative groups is used in many fields of mathematics, physics, and engineering. Nowadays, this method finds increasingly wide application to non-commutative groups. Along with the one-sided convolution operators and the corresponding convolution equations, two-sided convolution operators on noncommutative groups are studied. Two-sided convolution operators have a number of applications in complex analysis and are used in quantum mechanics. In this paper, two-sided convolutions on arbitrary finite noncommutative groups are considered. A criterion for the inversibility of the two-sided convolution operator is obtained. An algorithm for solving the two-sided convolution equation on an arbitrary finite noncommutative group, using the Fourier transform, is developed. Estimates of the computational complexity of the algorithm developed are given. It is shown that the complexity of solving the two-sided convolution equation depends both on the type of the group representation and on the computational complexity of the Fourier transform. The algorithm is considered in detail on the example of the finite dihedral group $\mathbb{D}_m$ and the Heisenberg group $\mathbb{H}(\mathbb{F}_p)$ over a simple Galois field, and the results of numerical experiments are presented.

Key words: two-sided convolution operators, two-sided convolution equations, fast Fourier transform, finite noncommutative groups, finite Heisenberg group, dihedral group.

DOI: https://doi.org/10.31857/S004446690003582-1

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English version:
Computational Mathematics and Mathematical Physics, 2018, 58:10, 1562–1572

Bibliographic databases:

UDC: 517.926
Revised: 15.03.2018

Citation: V. M. Deundyak, D. A. Leonov, “Fourier method for solving two-sided convolution equations on finite noncommutative groups”, Zh. Vychisl. Mat. Mat. Fiz., 58:10 (2018), 1616–1626; Comput. Math. Math. Phys., 58:10 (2018), 1562–1572

Citation in format AMSBIB
\Bibitem{DeuLeo18} \by V.~M.~Deundyak, D.~A.~Leonov \paper Fourier method for solving two-sided convolution equations on finite noncommutative groups \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2018 \vol 58 \issue 10 \pages 1616--1626 \mathnet{http://mi.mathnet.ru/zvmmf10789} \crossref{https://doi.org/10.31857/S004446690003582-1} \elib{https://elibrary.ru/item.asp?id=36715788} \transl \jour Comput. Math. Math. Phys. \yr 2018 \vol 58 \issue 10 \pages 1562--1572 \crossref{https://doi.org/10.1134/S0965542518100044} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000449497100003} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85056106862} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. V. Denisenko, V. M. Deundyak, “Fredholm Property of Integral Operators with Homogeneous Kernels of Compact Type in the $L_2$ Space on the Heisenberg Group”, Proc. Steklov Inst. Math., 308 (2020), 155–167