Kolmogorov Width of Classes of Smooth Functions on the Sphere |
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Authors: | Gavin Brown Dai Feng Sun Yong Sheng |
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Affiliation: | University of Sydney, Sydney, NSW 2006, Australiaf1;Department of Mathematics, Beijing Normal University, Beijing, 100875, Chinaf2f3 |
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Abstract: | Let d−1{(x1,…,xd)d:x21+···+x2d=1} be the unit sphere of the d-dimensional Euclidean space d. For r>0, we denote by Brp (1p∞) the class of functions f on d−1 representable in the formwhere dσ(y) denotes the usual Lebesgue measure on d−1, and Pλk(t) is the ultraspherical polynomial.For 1p,q∞, the Kolmogorov N-width of Brp in Lq(d−1) is given bythe left-most infimum being taken over all N-dimensional subspaces XN of Lq(d−1).The main result in this paper is that for r2(d−1)2,where ANBN means that there exists a positive constant C, independent of N, such that C−1ANBNCAN.This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions. |
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Keywords: | Marcinkiewicz– Zygmund inequality spherical harmonics Kolmogorov width weighted Kashin-type inequality. |
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