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Abstract and Applied Analysis
Volume 2013, Article ID 154637, 9 pages
http://dx.doi.org/10.1155/2013/154637
Research Article

Uniform Bounds of Aliasing and Truncated Errors in Sampling Series of Functions from Anisotropic Besov Class

School of Mathematics and LPMC, Nankai University, Tianjin 300071, China

Received 1 May 2013; Accepted 11 June 2013

Academic Editor: Yiming Ying

Copyright © 2013 Peixin Ye and Yongjie Han. 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.

Abstract

Errors appear when the Shannon sampling series is applied to approximate a signal in real life. This is because a signal may not be bandlimited, the sampling series may have to be truncated, and the sampled values may not be exact and may have to be quantized. In this paper, we truncate the multidimensional Shannon sampling series via localized sampling and obtain the uniform bounds of aliasing and truncation errors for functions from anisotropic Besov class without any decay assumption. The bounds are optimal up to a logarithmic factor. Moreover, we derive the corresponding results for the case that the sampled values are given by a linear functional and its integer translations. Finally we give some applications.