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Non-uniform Randomized Sampling for Multivariate Approximation by High Order Parzen Windows

Published online by Cambridge University Press:  20 November 2018

Xiang-Jun Zhou ,
Lei Shi  and
Ding-Xuan Zhou
Xiang-Jun Zhou
Affiliation:
Joint Advanced Research Center in Suzhou, University of Science and Technology of China and City University of Hong Kong, Suzhou, Jiangshu, 215123, China e-mail: 50009229@student.cityu.edu.hksl1983@mail.ustc.edu.cn
Lei Shi
Affiliation:
Joint Advanced Research Center in Suzhou, University of Science and Technology of China and City University of Hong Kong, Suzhou, Jiangshu, 215123, China e-mail: 50009229@student.cityu.edu.hksl1983@mail.ustc.edu.cn
Ding-Xuan Zhou
Affiliation:
Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China e-mail: mazhou@cityu.edu.hk
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Abstract

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We consider approximation of multivariate functions in Sobolev spaces by high order Parzen windows in a non-uniform sampling setting. Sampling points are neither i.i.d. nor regular, but are noised from regular grids by non-uniform shifts of a probability density function. Sample function values at sampling points are drawn according to probability measures with expected values being values of the approximated function. The approximation orders are estimated by means of regularity of the approximated function, the density function, and the order of the Parzen windows, under suitable choices of the scaling parameter.

Keywords

68T0562J02multivariate approximationSobolev spacesnon-uniform randomized samplinghigh order Parzen windowsconvergence rates
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 3 , 01 September 2011 , pp. 566 - 576
Copyright
Copyright © Canadian Mathematical Society 2011

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