N.I. Chernykh. Periodic wavelets on a multidimensional sphere and their application for function approximation ... P. 255-267

The author's scheme for constructing a multiresolution analysis on a sphere in $\mathbb R^3$ with respect to the spherical coordinates, which was published in 2019, is extended to spheres in $\mathbb R^n$ $(n\ge 3)$. In contrast to other papers, only periodic wavelets on the axis and their tensor products are used. Approximation properties are studied only for the wavelets based on the simplest scalar wavelets of Kotel'nikov-Meyer type with the compact support of their Fourier transforms. The implementation of the idea of a smooth continuation of functions from a sphere to $2\pi$-periodic functions in the polar coordinates analytically (without the complicated geometric interpretation made by the author earlier in $\mathbb R^3$) turned out to be very simple.

Keywords: wavelet, scaling function, approximation

Received September 28, 2020

Revised November 4, 2020

Accepted November 16, 2020

Funding Agency: This study is a part of the research carried out at the Ural Mathematical Center and was supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Nikolai Ivanovich Chernykh, Dr. Phys.-Math. Sci., Prof., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: chernykh@imm.uran.ru

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Cite this article as: N.I. Chernykh. Periodic wavelets on a multidimensional sphere and their application for function approximation, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 4, pp. 255–267.