On Filtered Polynomial Approximation on the Sphere

On Filtered Polynomial Approximation on the Sphere This paper considers filtered polynomial approximations on the unit sphere $$\mathbb {S}^d\subset \mathbb {R}^{d+1}$$ S d ⊂ R d + 1 , obtained by truncating smoothly the Fourier series of an integrable function f with the help of a “filter” h, which is a real-valued continuous function on $$[0,\infty )$$ [ 0 , ∞ ) such that $$h(t)=1$$ h ( t ) = 1 for $$t\in [0,1]$$ t ∈ [ 0 , 1 ] and $$h(t)=0$$ h ( t ) = 0 for $$t\ge 2$$ t ≥ 2 . The resulting “filtered polynomial approximation” (a spherical polynomial of degree $$2L-1$$ 2 L - 1 ) is then made fully discrete by approximating the inner product integrals by an N-point cubature rule of suitably high polynomial degree of precision, giving an approximation called “filtered hyperinterpolation”. In this paper we require that the filter h and all its derivatives up to $$\lfloor \tfrac{d-1}{2}\rfloor $$ ⌊ d - 1 2 ⌋ are absolutely continuous, while its right and left derivatives of order $$\lfloor \tfrac{d+1}{2}\rfloor $$ ⌊ d + 1 2 ⌋ exist everywhere and are of bounded variation. Under this assumption we show that for a function f in the Sobolev space $$W^s_p(\mathbb {S}^d),\ 1\le p\le \infty $$ W p s ( S d ) , 1 ≤ p ≤ ∞ , both approximations are of the optimal order $$ L^{-s}$$ L - s , in the first case for $$s>0$$ s > 0 and in the second fully discrete case for $$s>d/p$$ s > d / p , conditions which in both cases cannot be weakened. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Fourier Analysis and Applications Springer Journals

On Filtered Polynomial Approximation on the Sphere

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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Fourier Analysis; Signal,Image and Speech Processing; Abstract Harmonic Analysis; Approximations and Expansions; Partial Differential Equations; Mathematical Methods in Physics
ISSN
1069-5869
eISSN
1531-5851
D.O.I.
10.1007/s00041-016-9493-7
Publisher site
See Article on Publisher Site

Abstract

This paper considers filtered polynomial approximations on the unit sphere $$\mathbb {S}^d\subset \mathbb {R}^{d+1}$$ S d ⊂ R d + 1 , obtained by truncating smoothly the Fourier series of an integrable function f with the help of a “filter” h, which is a real-valued continuous function on $$[0,\infty )$$ [ 0 , ∞ ) such that $$h(t)=1$$ h ( t ) = 1 for $$t\in [0,1]$$ t ∈ [ 0 , 1 ] and $$h(t)=0$$ h ( t ) = 0 for $$t\ge 2$$ t ≥ 2 . The resulting “filtered polynomial approximation” (a spherical polynomial of degree $$2L-1$$ 2 L - 1 ) is then made fully discrete by approximating the inner product integrals by an N-point cubature rule of suitably high polynomial degree of precision, giving an approximation called “filtered hyperinterpolation”. In this paper we require that the filter h and all its derivatives up to $$\lfloor \tfrac{d-1}{2}\rfloor $$ ⌊ d - 1 2 ⌋ are absolutely continuous, while its right and left derivatives of order $$\lfloor \tfrac{d+1}{2}\rfloor $$ ⌊ d + 1 2 ⌋ exist everywhere and are of bounded variation. Under this assumption we show that for a function f in the Sobolev space $$W^s_p(\mathbb {S}^d),\ 1\le p\le \infty $$ W p s ( S d ) , 1 ≤ p ≤ ∞ , both approximations are of the optimal order $$ L^{-s}$$ L - s , in the first case for $$s>0$$ s > 0 and in the second fully discrete case for $$s>d/p$$ s > d / p , conditions which in both cases cannot be weakened.

Journal

Journal of Fourier Analysis and ApplicationsSpringer Journals

Published: Jul 26, 2016

References

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