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| SOME EXTENSIONS OF PALEY-WIENNER THEOREM |
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Citation： |
Liu Youming,Sun Qiyu,Huang Daren.SOME EXTENSIONS OF PALEY-WIENNER THEOREM[J].Chinese Annals of Mathematics B,1998,19(3):331~340 |
| Page view： 1052
Net amount： 651 |
Authors： |
Liu Youming; Sun Qiyu;Huang Daren |
Foundation： |
Project supported by the Science
Foundation of New Stars of Beijing. |
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| Abstract： |
The Shannon's sampling theorem has many extensions, two of
which are to wavelet subspaces of $L^2(R)$ and to $B^2_{\pi}=:\{f(x,y)\in L^2(
R^2),$ supp $\hat{f}\subseteq [-\pi,\pi]\times[-\pi,\pi]\}$, where supp$\hat{f}
$ denotes the support of the Fourier transform of a function $f$. In fact, the
Paley-Wienner theorem says that each $f$ in $B_{\pi}^2$ can be recovered
from its sampled values $\{f(x_n,y_m)\}_{n,m}$ if $(x_n, y_m)$ satisfies
$|x_n-n|\leq L<\frac{1}{4}$ and $|y_m-m|\leq L<\frac{1}{4}$. Unfortunately this
theorem requires strongly the product structure of sampling set $\{(x_n, y_m)\}
_{m,n\in Z}$. This paper gives a sampling theorem in which the
sampling set has a general form $\{(x_{nm}, y_{nm})\}$. In addition, G.
Walter's sampling theorem is extended to wavelet subspaces of $L^2(R^2)$
and irregular sampling with the general sampling set $\{(x_{nm},y_{nm})\}$
is considered in the same spaces. All results in this work can be written
similarly in $n$-dimensional case for $n\geq 2$. |
Keywords： |
Sampling, Paley-Wienner theorem, Wavelets |
Classification： |
94A11, 42C15 |
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