Constructive Polynomial Approximation on the Sphere

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In this Subject: <a title="Mathematics and Statistics" href="/content/subcat?j_subject=195&j_availability=">Mathematics and Statistics
By this author: Sloan I.H. ;&nbsp; Womersley R.S.

Authors: Sloan I.H.; Womersley R.S.

Source: Journal of Approximation Theory, Volume 103, Number 1, March 2000 , pp. 91-118(28)

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Abstract:

This paper considers the problem of constructive approximation of a continuous function on the unit sphere S^r-1 sube Ropf ^r by a spherical polynomial from the space Popf _n of all spherical polynomials of degree les n. In particular, for r=3 it is shown that the hyperinterpolation approximation L_nf (in which the Fourier coefficients in the exact L₂ orthogonal projection P_nf are approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree les 2n) has the exact order Verbar L_n Verbar asymp n^1/2 for its uniform norm, provided the underlying quadrature rule satisfies an additional “quadrature regularity” assumption. For r=3, this rate of growth is the same as that of Verbar P_n Verbar , and is known to be optimal among all linear projections on Popf _n. For r ges 3 an upper bound on Verbar L_n Verbar of non-optimal asymptotic order O(n^(r-1)/2) also holds, without any special assumption on the quadrature rule. Copyright 2000 Academic Press.