Approximation of Bivariate Functions via Smooth Extensions
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...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | The Scientific World Journal |
| Online Access: | http://dx.doi.org/10.1155/2014/102062 |
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| _version_ | 1832559982229323776 |
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| author | Zhihua Zhang |
| author_facet | Zhihua Zhang |
| author_sort | Zhihua Zhang |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-8187b003559341bea19d63d8930085e1 |
| institution | Kabale University |
| issn | 2356-6140 1537-744X |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | The Scientific World Journal |
| spelling | doaj-art-8187b003559341bea19d63d8930085e12025-02-03T01:28:41ZengWileyThe Scientific World Journal2356-61401537-744X2014-01-01201410.1155/2014/102062102062Approximation of Bivariate Functions via Smooth ExtensionsZhihua Zhang0College of Global Change and Earth System Science, Beijing Normal University, Beijing 100875, ChinaFor 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.http://dx.doi.org/10.1155/2014/102062 |
| spellingShingle | Zhihua Zhang Approximation of Bivariate Functions via Smooth Extensions The Scientific World Journal |
| title | Approximation of Bivariate Functions via Smooth Extensions |
| title_full | Approximation of Bivariate Functions via Smooth Extensions |
| title_fullStr | Approximation of Bivariate Functions via Smooth Extensions |
| title_full_unstemmed | Approximation of Bivariate Functions via Smooth Extensions |
| title_short | Approximation of Bivariate Functions via Smooth Extensions |
| title_sort | approximation of bivariate functions via smooth extensions |
| url | http://dx.doi.org/10.1155/2014/102062 |
| work_keys_str_mv | AT zhihuazhang approximationofbivariatefunctionsviasmoothextensions |