Probabilistic and Average Widths of Sobolev Spaces on Compact Two-Point Homogeneous Spaces Equipped with a Gaussian Measure

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 \(1\le q\le \infty \) 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 \(L_q \ (1\le q<\infty )\) metric, and the Fourier partial summation operators are the asymptotic optimal linear operators in the \(L_q\ (1\le q\le \infty )\) metric and are (modulo a constant) as good as optimal nonlinear operators in the \(L_q \ (1\le q<\infty )\) metric.