Abstract
In this paper we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere S r−1 ⊂ Rr. The hyperinterpolation approximation L n ƒ, where ƒ ∈ C(S r −1), is derived from the exact L 2 orthogonal projection Π ƒ onto the space P r n (S r −1) of spherical polynomials of degree n or less, with the Fourier coefficients approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree ≤ 2n. We extend to arbitrary r the recent r = 3 result of Sloan and Womersley [9], by proving that under an additional “quadrature regularity” assumption on the quadrature rule, the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere is O(n r/2−1), which is the same as that of the orthogonal projection Πn, and best possible among all linear projections onto P r n (S r −1).
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References
A. N. Kolmogorov, S. V. Fomin (1970): Introductory Real Analysis. Englewood Cliffs, NJ: Prentice-Hall.
C. Müller (1966): Spherical Harmonics. Lecture Notes in Mathematics, Vol. 17. New York: Springer-Verlag.
C. Müller (1998): Analysis of Spherical Symmetries in Euclidean Spaces. Applied Mathematical Sciences, Vol. 129. New York: Springer-Verlag.
J. G. Ratcliffe (1994): Foundations of Hyperbolic Manifolds. Graduate Texts in Mathematics. New York: Springer-Verlag.
M. Reimer (1990). Constructive Theory of Multivariate Functions. Mannheim: Wissenschaftsverlag.
M. Reimer (to appear): Hyperinterpolation on the sphere at the minimal projection order. Preprint.
I. H. Sloan (1995): Polynomial interpolation and hyperinterpolation over general regions. J. Approx. Theory, 83:238–254.
I. H. Sloan (1997): Interpolation and hyperinterpolation on the sphere. In: Multivariate Approximation: Recent Trends and Results (W. Hausmann, K. Jetter, M. Reimer, eds.). Berlin: Academie Verlag (Wiley-VCH), pp. 255–268.
I. H. Sloan, R. Womersley (2000): Constructive polynomial approximation on the sphere. J. Approx. Theory, 103:91–118.
G. Szegó (1939): Orthogonal Polynomials. Providence, RI: American Mathematical Society.
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Communicated by E. B. Saff.
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Le Gia, T., Sloan, I.H. The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions. Constr. Approx 17, 249–265 (2001). https://doi.org/10.1007/s003650010025
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DOI: https://doi.org/10.1007/s003650010025