The Shannon's sampling theorem has many extensions, two of which are to wavelet subspaces of L 2(R) and to B 2 π =:{f(x,y)∈ L 2(R 2), supp ×} , where supp denotes the support of the...The Shannon's sampling theorem has many extensions, two of which are to wavelet subspaces of L 2(R) and to B 2 π =:{f(x,y)∈ L 2(R 2), supp ×} , where supp denotes the support of the Fourier transform of a function f . In fact, the Paley Wienner theorem says that each f in B 2 π can be recovered from its sampled values {f(x n,y m)} n,m if (x n, y m) satisfies |x n-n|L<14 and |y m-m| L<14 . Unfortunately this theorem requires strongly the product structure of sampling set {(x n, y m)} m,n∈ 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 n2 .展开更多
文摘The Shannon's sampling theorem has many extensions, two of which are to wavelet subspaces of L 2(R) and to B 2 π =:{f(x,y)∈ L 2(R 2), supp ×} , where supp denotes the support of the Fourier transform of a function f . In fact, the Paley Wienner theorem says that each f in B 2 π can be recovered from its sampled values {f(x n,y m)} n,m if (x n, y m) satisfies |x n-n|L<14 and |y m-m| L<14 . Unfortunately this theorem requires strongly the product structure of sampling set {(x n, y m)} m,n∈ 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 n2 .
