Spherical random sampling of localized functions on 𝕊ⁿ⁻¹

Abstract

In this paper, we show that the set of random points uniformly distributed over S n − 1 \mathbb {S}^{n-1} is a stable set of sampling for a class of localized functions in L 2 ( S n − 1 ) L^2(\mathbb {S}^{n-1}) with overwhelming probability. The probability bound of the random sampling inequality for the localized function is obtained from the matrix Bernstein inequality on the independent random matrices.