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DPR; Ukraine Donetsk
Year
2022
Volume
32
Issue
1
Pages
3-17
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Section Mathematics
Title Vector fields with zero flux through circles of fixed radius on $\mathbb{H}^2$
Author(-s) Volchkova N.P.a, Volchkov V.V.b
Affiliations Donetsk National Technical Universitya, Donetsk National Universityb
Abstract A classic property of a periodic function on the real axis is the possibility of its representation by a trigonometric Fourier series. The natural analogue of the periodicity condition in Euclidean space $\mathbb{R}^m$ is the constancy of integrals of a function over all balls (or spheres) of fixed radius. Functions with the indicated property can be expanded in a Fourier series in terms of spherical harmonics whose coefficients are expanded into series in Bessel functions. This fact can be generalized to vector fields in $\mathbb{R}^m$ with zero flux through spheres of fixed radius. In this paper we study vector fields which have zero flux through every circle of fixed radius on the Lobachevskii plane $\mathbb{H}^2$. A description of such fields in the form of series in terms of hypergeometric functions is obtained. These results can be used to solve problems concerning harmonic analysis of vector fields on domains in $\mathbb{H}^2$.
Keywords vector fields, Lobachevskii plane, zero spherical means, Horn hypergeometric series
UDC 517.588
MSC 53C65, 44A35
DOI 10.35634/vm220101
Received 28 January 2022
Language Russian
Citation Volchkova N.P., Volchkov V.V. Vector fields with zero flux through circles of fixed radius on $\mathbb{H}^2$, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 1, pp. 3-17.
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