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 Mat. Zametki, 2003, Volume 74, Issue 2, Pages 292–300 (Mi mz259)

Polynomial Wavelet-Type Expansions on the Sphere

A. Askari-Hemmata, M. A. Dehghana, M. A. Skopinab

a Valiasr University
b Saint-Petersburg State University

Abstract: We present a polynomial wavelet-type system on $S^d$ such that any continuous function can be expanded with respect to these “wavelets”. The order of the growth of the degrees of the polynomials is optimal. The coefficients in the expansion are the inner products of the function and the corresponding element of a “dual wavelet system”. The “dual wavelets system” is also a polynomial system with the same growth of degrees of polynomials. The system is redundant. A construction of a polynomial basis is also presented. In contrast to our wavelet-type system, this basis is not suitable for implementation, because, first, there are no explicit formulas for the coefficient functionals and, second, the growth of the degrees of polynomials is too rapid.

DOI: https://doi.org/10.4213/mzm259

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English version:
Mathematical Notes, 2003, 74:2, 278–285

Bibliographic databases:

UDC: 517.5

Citation: A. Askari-Hemmat, M. A. Dehghan, M. A. Skopina, “Polynomial Wavelet-Type Expansions on the Sphere”, Mat. Zametki, 74:2 (2003), 292–300; Math. Notes, 74:2 (2003), 278–285

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz259
• https://doi.org/10.4213/mzm259
• http://mi.mathnet.ru/eng/mz/v74/i2/p292

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This publication is cited in the following articles:
1. Fernandez NL, Prestin J, “Interpolatory band-limited wavelet bases on the sphere”, Constructive Approximation, 23:1 (2006), 79–101
2. Feng, D, “Characterizations of function spaces on the sphere using frames”, Transactions of the American Mathematical Society, 359:2 (2007), 567
3. T. G. zhao, L. Naing, W. X. Yue, “Some New Features of the Boubaker Polynomials Expansion Scheme BPES”, Math. Notes, 87:2 (2010), 165–168
4. Arfaoui S., Rezgui I., BenMabrouk A., “Wavelet Analysis on the Sphere: Spheroidal Wavelets”, Wavelet Analysis on the Sphere: Spheroidal Wavelets, Walter de Gruyter Gmbh, 2017, 1–144
5. Nikolai I. Chernykh, “Interpolating wavelets on the sphere”, Ural Math. J., 5:2 (2019), 3–12
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