## Abstract and Applied Analysis

### Analysis of Approximation by Linear Operators on Variable ${L}_{\rho }^{p(·)}$ 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.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014 (2014), Article ID 454375, 10 pages.

Dates
First available in Project Euclid: 2 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412277113

Digital Object Identifier
doi:10.1155/2014/454375

Mathematical Reviews number (MathSciNet)
MR3240539

Zentralblatt MATH identifier
07022406

#### Citation

Li, Bing-Zheng; Zhou, Ding-Xuan. Analysis of Approximation by Linear Operators on Variable ${L}_{\rho }^{p(·)}$ Spaces and Applications in Learning Theory. Abstr. Appl. Anal. 2014 (2014), Article ID 454375, 10 pages. doi:10.1155/2014/454375. https://projecteuclid.org/euclid.aaa/1412277113