INTERPOLATING WAVELETS ON THE SPHERE

Nikolai I. Chernykh     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620990; Ural Federal University, 51 Lenin aven., Ekaterinburg, 620000, Russian Federation)

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

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References


  1. Arfaoui S., Rezgui I., Mabrouk A.B. Wavelet Analysis on the Sphere: Spheroidal Wavelets. Berlin: Walter de Gruyter GmbH & Co KG, 2017. 144 p.
  2. Askari-Hemmat A., Dehghan M.A., Skopina M. Polynomial Wavelet-Type Expansions on the Sphere. Math. Notes, 2003. Vol. 74, No. 2. P. 278–285. DOI: 10.1023/A:1025016510773
  3. Chernykh N.I., Subbotin Yu.N. Interpolating-orthogonal wavelet systems. Proc. Steklov Inst. Math., 2009. Vol. 264, Suppl. 1. P. 107–115. DOI: 10.1134/S0081543809050083
  4. Dahlke S., Dahmen W., Weinreich I., Schmitt E. Multiresolution analysis and wavelets on \(\mathbb{S}^2\) and \(\mathbb{S}^3\). Numer. Funct. Anal. Optim., 1995. Vol. 16, No. 1–2. P. 19–41. DOI: 10.1080/01630569508816605
  5. Dai F. Characterizations of function spaces on the sphere using frames. Trans. Amer. Math. Soc., 2007. Vol. 359, No. 2. P. 567–589. DOI: 10.1090/S0002-9947-06-04030-X
  6. Farkov Yu. B-spline wavelets on the sphere. In: Proc. of the Intern. Workshop “Self-Similar Systems”, 1999. Vol. 30. P. 79–82.
  7. Freeden W., Schreiner M. Orthogonal and nonorthogonal multiresolution analysis, scale discrete and exact fully discrete wavelet transform on the sphere. Constr. Approx., 1998. Vol. 14, No. 4. P. 493–515. DOI: 10.1007/s003659900087
  8. Schröder P., Sweldens W. Spherical wavelets: Efficiently representing functions on the sphere. In: Wavelets in the Geosciences. Lect. Notes in Earth Sci., vol. 90. 1995. P. 158–188. DOI: 10.1007/BFb0011096
  9. Skopina M. Polynomial Expansions of Continuous Functions on the Sphere and on the Disk. IMI Research Reports, Department of Mathematics, University of South Carolina, 2001. Preprint, Vol. 5. 13 p. DOI: http://imi.cas.sc.edu/django/site media/media/papers/2001/2001 05.pdf
  10. Subbotin Yu.N., Chernykh N.I. Interpolation wavelets in boundary value problems. Proc. Steklov Inst. Math., 2018. Vol. 300, Suppl. 1. P. 172–183. DOI: 10.1134/S0081543818020177



DOI: http://dx.doi.org/10.15826/umj.2019.2.001

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