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Abstract and Applied Analysis
Volume 2014, Article ID 454375, 10 pages
Research Article

Analysis of Approximation by Linear Operators on Variable Spaces and Applications in Learning Theory

1Department of Mathematics, Zhejiang University, Hangzhou 310027, China
2Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received 7 May 2014; Accepted 30 June 2014; Published 16 July 2014

Academic Editor: Uno Hämarik

Copyright © 2014 Bing-Zheng Li and Ding-Xuan Zhou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper is concerned with approximation on variable spaces associated with a general exponent function and a general bounded Borel measure on an open subset of . We mainly consider approximation by Bernstein type linear operators. Under an assumption of log-Hölder continuity of the exponent function , we verify a conjecture raised previously about the uniform boundedness of Bernstein-Durrmeyer and Bernstein-Kantorovich operators on the 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.