Analysis of Approximation by Linear Operators on Variable Lρp(·) Spaces and Applications in Learning Theory

Abstract

This paper is concerned with approximation on variable $L_{\rho }^{p(·)}$ spaces associated with a general exponent function $p$ and a general bounded Borel measure $\rho$ on an open subset $\mathrm{\Omega }$ of ${\mathbb{R}}^d$. We mainly consider approximation by Bernstein type linear operators. Under an assumption of log-Hölder continuity of the exponent function $p$, we verify a conjecture raised previously about the uniform boundedness of Bernstein-Durrmeyer and Bernstein-Kantorovich operators on the $L_{\rho }^{p(·)}$ space. Quantitative estimates for the approximation are provided for high orders of approximation by linear combinations of such positive linear operators. Motivating connections to classification and quantile regression problems in learning theory are also described.