Volume 19, pp. 84-93, 2005.

Localized polynomial bases on the sphere

Noemí Laín Fernández

Abstract

The subject of many areas of investigation, such as meteorology or crystallography, is the reconstruction of a continuous signal on the $2$-sphere from scattered data. A classical approximation method is polynomial interpolation. Let $V_n$ denote the space of polynomials of degree at most $n$ on the unit sphere ${\mathbf{S}}^2\subset {\mathbf{R}}^3$. As it is well known, the so-called spherical harmonics form an orthonormal basis of the space $V_n$. Since these functions exhibit a poor localization behavior, it is natural to ask for better localized bases. Given $\{{\xi }_i{\}}_{i=1,\dots ,(n+1{)}^2}\subset {\mathbf{S}}^2$, we consider the spherical polynomials $${\phi }_i^n(\xi ):=\sum _{l=0}^n\frac{2l+1}{4\pi }{P}_l({\xi }_i\cdot \xi ),$$ where $P_l$ denotes the Legendre polynomial of degree $l$ normalized according to the condition $P_l(1)=1$. In this paper, we present systems of $(n+1{)}^2$ points on ${\mathbf{S}}^2$ that yield localized polynomial bases of the above form.

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Key words

fundamental systems, localization, matrix condition, reproducing kernel.

AMS subject classifications

41A05, 65D05, 15A12.

ETNA articles which cite this article

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