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

, Volume 23 (4) – Jul 26, 2016
14 pages

/lp/springer_journal/on-filtered-polynomial-approximation-on-the-sphere-Q5cdDXeZYS
Publisher
Springer US
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

DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month Explore the DeepDyve Library Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve Freelancer DeepDyve Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations