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 Ural Math. J., 2019, Volume 5, Issue 2, Pages 3–12 (Mi umj98)

Interpolating wavelets on the sphere

Nikolai I. Chernykhab

a Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya str., Ekaterinburg, Russia, 620990
b Ural Federal University, 51 Lenin aven., Ekaterinburg, 620000, Russia

Abstract: There are several works where bases of wavelets on the sphere (mainly orthogonal and wavelet-like bases) were constructed. In all such constructions, the authors seek to preserve the most important properties of classical wavelets including constructions on the basis of the lifting-scheme. In the present paper, we propose one more construction of wavelets on the sphere. Although two of three systems of wavelets constructed in this paper are orthogonal, we are more interested in their interpolation properties. Our main idea consists in a special double expansion of the unit sphere in $\mathbb{R}^3$ such that any continuous function on this sphere defined in spherical coordinates is easily mapped into a $2\pi$-periodic function on the plane. After that everything becomes simple, since the classical scheme of the tensor product of one-dimensional bases of functional spaces works to construct bases of spaces of functions of several variables.

Keywords: Wavelets, Multiresolution analysis, Scaling functions, Interpolating wavelets, Best approximation, Trigonometric polynomials.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 02.A03.21.0006 This work 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).

DOI: https://doi.org/10.15826/umj.2019.2.001

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Citation: Nikolai I. Chernykh, “Interpolating wavelets on the sphere”, Ural Math. J., 5:2 (2019), 3–12

Citation in format AMSBIB
\Bibitem{Che19} \by Nikolai I. Chernykh \paper Interpolating wavelets on the sphere \jour Ural Math. J. \yr 2019 \vol 5 \issue 2 \pages 3--12 \mathnet{http://mi.mathnet.ru/umj98} \crossref{https://doi.org/10.15826/umj.2019.2.001} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=MR4054646} \zmath{https://zbmath.org/?q=an:07255674} \elib{https://elibrary.ru/item.asp?id=41672789} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85078782898} 

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This publication is cited in the following articles:
1. N. I. Chernykh, “Periodicheskie vspleski na mnogomernoi sfere i ikh primenenie dlya approksimatsii funktsii”, Tr. IMM UrO RAN, 26, no. 4, 2020, 255–267
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