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- W2037295557 abstract "Let S d −1 ≔{( x 1 ,…, x d )∈ R d : x 2 1 +···+ x 2 d =1} be the unit sphere of the d -dimensional Euclidean space R d . For r >0, we denote by B r p (1⩽ p ⩽∞) the class of functions f on S d −1 representable in the formwhere dσ ( y ) denotes the usual Lebesgue measure on S d −1 , λ= d−2 2 and P λ k ( t ) is the ultraspherical polynomial. For 1⩽ p,q ⩽∞, the Kolmogorov N -width of B r p in L q ( S d −1 ) is given bythe left-most infimum being taken over all N -dimensional subspaces X N of L q ( S d −1 ). The main result in this paper is that for r ⩾2( d −1) 2 ,where A N ≍ B N means that there exists a positive constant C , independent of N , such that C −1 A N ⩽ B N ⩽ CA N . This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions." @default.
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- W2037295557 date "2002-12-01" @default.
- W2037295557 modified "2023-09-30" @default.
- W2037295557 title "Kolmogorov Width of Classes of Smooth Functions on the Sphere Sd−1" @default.
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- W2037295557 doi "https://doi.org/10.1006/jcom.2002.0656" @default.
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