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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
C. Foias and I. Singer, “Some remarks on strongly independent sequences and bases in Banach spaces,” Rev. Math. Pure et Appl. Acad. R.P.R., 6 (1961), no. 3, 589–594.
Al. A. Privalov, “On the growth of degrees of polynomial bases and approximation of trigonometric projections,” Mat. Zametki [Math. Notes], 42 (1987), no. 2, 207–214.
R. A. Lorentz and A. A. Saakyan (Sahakian), “Orthogonal trigonometric Schauder bases of optimal degree for C(0, 2π),” J. Fourier Anal. Appl., 1 (1994), no. 1, 103–112.
M. A. Skopina, “Orthogonal polynomial Schauder bases for C[-1, 1] of optimal degree,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 192 (2001), no. 3, 115–136.
W. Freeden and M. Schreiner, “Orthogonal and non-orthogonal multiresolution analysis, scale discrete and exact fully discrete wavelet transform on the sphere,” Constructive Approximation, 14 (1998), 493–515.
Yu. Farkov, “B-spline wavelets on the sphere,” in: “Self-Similar Systems,” Proceedings of the International Workshop (July 30–August 7, 1998, Dubna, Russia), JINR, E5-99-38, Dubna, 1999, pp. 79–82.
M. Skopina, “Polynomial expansions of continuous functions on the sphere and on the disk,” in: Preprint no. 2001:5, University of South Carolina, Department of Mathematics, 2001.
C. Müller, Spherical Harmonics, Lecture Notes in Math., vol. 17, Springer-Verlag, Berlin, 1966.
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, NJ, 1971.
H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, “Spherical Marcinkiewicz–Zygmund inequalities and positive quadrature,” Math. Comp., 70 (2001), no. 235, 1113–1130.
H. N. Mhaskar, F. J. Narcowich, J. D. Ward, and J. Prestin, “Polynomial frames on the sphere,” Adv. Comput. Math., 13 (2000), 387–403.
R. Askey, Orthogonal Polynomials and Spherical Functions, SIAM, Philadelphia, 1975.
G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., New York, 1959.
B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Izd. AFTs, Moscow, 1999.
Wang Kunyang and Li Luoking, Harmonic Analysis and Approximation on the Unit Sphere. Graduate Series in Mathematics, 2000.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Askari-Hemmat, A., Dehghan, M.A. & Skopina, M. Polynomial Wavelet-Type Expansions on the Sphere. Mathematical Notes 74, 278–285 (2003). https://doi.org/10.1023/A:1025016510773
Issue Date:
DOI: https://doi.org/10.1023/A:1025016510773