Special Issue: Positive Polynomial Sums, Orthogonal Expansions and ApplicationsNo Access

Approximation of functions on the sphere on a Sobolev space with a Gaussian measure in the probabilistic case setting

Heping Wang

School of Mathematical Sciences, BCMIIS, Capital Normal University, Beijing 100048, P. R. China

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and

Xuebo Zhai

School of Mathematics and Statistics, Zaozhuang University, Zaozhuang 277160, Shandong, P. R. China

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https://doi.org/10.1142/S0219691314610128Cited by:2 (Source: Crossref)

Abstract

In this paper, we discuss the best approximation of functions on the sphere by spherical polynomials and the approximation by the Fourier partial summation operators and the Vallée-Poussin operators, on a Sobolev space with a Gaussian measure in the probabilistic case setting, and get the probabilistic error estimation. We show that in the probabilistic case setting, the Fourier partial summation operators and the Vallée-Poussin operators are the order optimal linear operators in the L_q space for 1 ≤ q ≤ ∞, but the spherical polynomial spaces are not order optimal in the L_q space for 2 < q ≤ ∞. This is completely different from the situation in the average case setting, which the spherical polynomial spaces are order optimal in the L_q space for 1 ≤ q < ∞. Also, in the L_q space for 1 ≤ q ≤ ∞, worst-case order optimal subspaces are also order optimal in the probabilistic case setting.

Dedicated to Professor Kunyang Wang on the occasion of his 70th birthday

Keywords:

AMSC: 41A25, 41A63, 42A61, 46C99

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