Uniform approximation on the sphere by least squares polynomials
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The paper concerns the uniform polynomial approximation of a function f, continuous on the unit Euclidean sphere of ℝ3 and known only at a finite number of points that are somehow uniformly distributed on the sphere. First, we focus on least squares polynomial approximation and prove that the related Lebesgue constants w.r.t. the uniform norm grow at the optimal rate. Then, we consider delayed arithmetic means of least squares polynomials whose degrees vary from n − m up to n + m, being m = θn for any fixed parameter 0 < θ < 1. As n tends to infinity, we prove that these polynomials uniformly converge to f at the near-best polynomial approximation rate. Moreover, for fixed n, by using the same data points, we can further improve the approximation by suitably modulating the action ray m determined by the parameter θ. Some numerical experiments are given to illustrate the theoretical results.
KeywordsPolynomial approximation on the sphere Least squares approximation Uniform approximation Lebesgue constant De la Vallée Poussin type mean
Mathematics Subject Classification (2010)41-A10 65-D99 33-C45
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The authors would like to thank the anonymous referee for his valuable comments and suggestions to improve the quality of the paper, and Ed Saff for providing us reference . The research of the first author was partially supported by GNCS–INDAM, and that of the second author by the Research Council KU Leuven, C1-project (Numerical Linear Algebra and Polynomial Computations), and by the Fund for Scientific Research–Flanders (Belgium), “SeLMA” - EOS reference number: 30468160.
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