Probabilistic and average widths of Sobolev spaces on compact two-point homogeneous spaces equipped with a Gaussian measure
type: Item
instance of: scholarly article
title: Probabilistic and average widths of Sobolev spaces on compact two-point homogeneous spaces equipped with a Gaussian measure
zbMATH Open document ID: 1310.41023
published in: Constructive Approximation
publication date: 2015-01-09T00:00:00Z
review text: Authors's abstract: Asymptotic values of the probabilistic Kolmogorov and linear \((n,\delta)\)-widths \(d_{n,\delta}(W_2^r(\mathbb {M}^{d-1})\), \(\mu ,L_q(\mathbb {M}^{d-1}))\) and \(\lambda _{n,\delta }(W_2^r(\mathbb {M}^{d-1})\), \(\mu ,L_q(\mathbb {M}^{d-1}))\), the \(p\)-average Kolmogorov and linear \(n\)-widths \(d_n^{(a)}(W_2^r(\mathbb {M}^{d-1})\), \(\mu ,L_q(\mathbb {M}^{d-1}))_p\) and \(\lambda _n^{(a)}(W_2^r(\mathbb {M}^{d-1})\), \(\mu ,L_q(\mathbb {M}^{d-1}))_p\) of the Sobolev space \(W_2^r(\mathbb {M}^{d-1})\) with the Gaussian measure \(\mu \) on compact two-point homogeneous spaces \(\mathbb {M}^{d-1}\) are determined for all \(1\leq q\leq \infty \) and \(0<p<\infty \). These quantities depend heavily on the smooth index of the Cameron-Martin space of the Gaussian measure \(\mu \). We also show that, in the average case setting with the average being with respect to the above measure \(\mu \), the spherical polynomial subspaces are the asymptotic optimal subspaces in the \(L_q \;(1\leq q<\infty)\) metric, and the Fourier partial summation operators are the asymptotic optimal linear operators in the \(L_q\;(1\leq q\leq \infty)\) metric and are (modulo a constant) as good as optimal nonlinear operators in the \(L_q \;(1\leq q<\infty)\) metric.
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 42A61
Mathematics Subject Classification ID: 46C99
zbMATH DE Number: 6385025
zbMATH Keywords: compact two-point homogeneous spaces
zbMATH Keywords: average widths
zbMATH Keywords: probabilistic widths
zbMATH Keywords: Sobolev space
zbMATH Keywords: Gaussian measure
author: Heping Wang
reviewed by: Yuri A. Farkov
MaRDI profile type: Publication
full work available at URL: s00365-014-9231-9
OpenAlex ID: W1986736351
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DOI: 10.1007/S00365-014-9231-9
description: scientific article; zbMATH DE number 6385025