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The Scientific World Journal
Volume 2014 (2014), Article ID 102062, 16 pages
http://dx.doi.org/10.1155/2014/102062
Research Article

Approximation of Bivariate Functions via Smooth Extensions

College of Global Change and Earth System Science, Beijing Normal University, Beijing 100875, China

Received 10 December 2013; Accepted 26 December 2013; Published 10 February 2014

Academic Editors: N. I. Mahmudov and S. Wang

Copyright © 2014 Zhihua Zhang. 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

For a smooth bivariate function defined on a general domain with arbitrary shape, it is difficult to do Fourier approximation or wavelet approximation. In order to solve these problems, in this paper, we give an extension of the bivariate function on a general domain with arbitrary shape to a smooth, periodic function in the whole space or to a smooth, compactly supported function in the whole space. These smooth extensions have simple and clear representations which are determined by this bivariate function and some polynomials. After that, we expand the smooth, periodic function into a Fourier series or a periodic wavelet series or we expand the smooth, compactly supported function into a wavelet series. Since our extensions are smooth, the obtained Fourier coefficients or wavelet coefficients decay very fast. Since our extension tools are polynomials, the moment theorem shows that a lot of wavelet coefficients vanish. From this, with the help of well-known approximation theorems, using our extension methods, the Fourier approximation and the wavelet approximation of the bivariate function on the general domain with small error are obtained.