Kolmogorov width of classes of smooth functions on the sphere Sd-1

• Authors: Gavin Brown;Dai Feng;Sun Yong Sheng
• Venue: Journal of Complexity
• Year: 2002

Let Sd-1 = {(x1,....,xd ∈ Rd : x12 + ...... + xd2 = 1} be the unit sphere of the d- dimensional Euclidean space Rd. For r Bpp (1 ≤ p ≤ ∞) the class of functions f on Sd-1 representable in the form f(x) = φ*Fr(x) = ∫Sd-1 φ(y)Fr(xy)dσ(y), ||φ||p ≤ 1, where dσ(y) denotes the usual Lebesgue measure on Sd-1, Fr(xy) = ∑ ∞ k=1 (k(k + 2λ))-r/2 γ(λ)(k+λ)/2πλ+1 Pkλ(xy), λ = d-2/2 and Pkλ(t) is the ultraspherical polynomial. For 1 ≤ p,q ≤ ∞, the Kolmogorov N-width of Bpr in Lq(Sd-1) is given by dN(Bpr, Lq) = XN supf∈Bprg∈XN||f - g||q, the left-most infimum being taken over all N-dimensional subspaces XN of Lq(Sd-1). The main result in this paper is that for r ≥ 2(d - 1)2, dN(Bpr, Lq) = = {N-r/d-1 if 2 ≤ p-r/d-1-1/p+1/2 if 1≤p≤q≤∞, where AN = NN means that there exists a positive constant C, independent of N, such that C-1 AN ≤ BN ≤ CAN. This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions.