## 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 ${\bf S}^2 \subset {\bf R}^3$. 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 $\lbrace\xi_i\rbrace_{i=1,\ldots,(n+1)^2}\subset {\bf S}^2$, we consider the spherical polynomials $\varphi_{i}^n(\xi):=\sum\limits_{l=0}^{n}\frac{2l+1}{4\pi}\,P_{l}(\xi_{i}\cdot\xi),$ 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 ${\bf S}^2$ 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.