关于L~2(R~d)中仿射子空间小波标架的一个注记
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摘要
研究了L~2(R~d)的有限生成仿射子空间中小波标架的构造.证明了任意有限生成仿射子空间都容许一个具有有限多个生成元的Parseval小波标架,并且得到了仿射子空间是约化子空间的一个充分条件.对其傅里叶变换是一个特征函数的单个函数生成的仿射子空间,得到了与小波标架构造相关的投影算子在傅里叶域上的明确表达式,同时也给出了一些例子.
This paper addresses the construction of wavelet frames in the setting of finitely generated affine subspaces of L~2(R~d).It is proved that an arbitrary finitely generated affine subspace admits a Parseval wavelet frame with finitely many generators.A sufficient condition is obtained for an affine subspace to be a reducing subspace.For a class of affine subspaces generated by a single function whose Fourier transform is a characteristic function,we derive an explicit Fourier-domain expression of the projection operators related to the construction of wavelet frames.Some examples are also provided.
This paper addresses the construction of wavelet frames in the setting of finitely generated affine subspaces of L~2(R~d).It is proved that an arbitrary finitely generated affine subspace admits a Parseval wavelet frame with finitely many generators.A sufficient condition is obtained for an affine subspace to be a reducing subspace.For a class of affine subspaces generated by a single function whose Fourier transform is a characteristic function,we derive an explicit Fourier-domain expression of the projection operators related to the construction of wavelet frames.Some examples are also provided.
引文
[1]Dai X,Diao Y,Gu Q,Han D.Frame wavelets in subspaces of L~2(R~d).Proc Amer Math Soc,2002,130: 3259-3267
[2]Zhou P Y,Li Y Z.Multivariate FMRAs and FMRA frame wavelets for reducing subspaces of L~2(R~d). Kyoto J Math,2010,50:83-99
[3]Weiss G,Wilson E N.The Mathematical Theory of Wavelets,in:Twentieth Century Harmonic Analysis-A Celebration//J S Byrnes(Ed).Proceedings of the NATO Advanced Study Institiute.Dordrecht:Kluwer Academic Publishers,2001:329-366
[4]Dai X,Diao Y,Gu Q.Subspaces with normalized tight frame wavelets in R.Proc Amer Math Soc,2002, 130:1661-1667
[5]Dai X,Diao Y,Gu Q,Han D.The existence of subspace wavelet sets.J Comput Appl Math,2003,155: 83-90
[6]Lian Q F,Li Y Z.Reducing subspace frame multiresolution analysis and frame wavelets.Commun Pure Appl Anal,2007,6:741-756
[7]Gu Q,Han D.Wavelet frames for(not necessarily reducing) affine subspaces.Appl Comput Harmon Anal, 2009,27:47-54
[8]Christensen O.An Introduction to Frames and Riesz Bases.Boston:Birkhauser,2003
[9]李云章,周凤英.L~2(R~d)中约化子空间上的仿射与伪仿射对偶小波标架.数学学报,2010,53:551-562
[10]de Boor C,DeVore R,Ron A.Approximation from shift-invariant subspaces of L~2(R~d).Trans Amer Math Soc,1994,341:787-806
[2]Zhou P Y,Li Y Z.Multivariate FMRAs and FMRA frame wavelets for reducing subspaces of L~2(R~d). Kyoto J Math,2010,50:83-99
[3]Weiss G,Wilson E N.The Mathematical Theory of Wavelets,in:Twentieth Century Harmonic Analysis-A Celebration//J S Byrnes(Ed).Proceedings of the NATO Advanced Study Institiute.Dordrecht:Kluwer Academic Publishers,2001:329-366
[4]Dai X,Diao Y,Gu Q.Subspaces with normalized tight frame wavelets in R.Proc Amer Math Soc,2002, 130:1661-1667
[5]Dai X,Diao Y,Gu Q,Han D.The existence of subspace wavelet sets.J Comput Appl Math,2003,155: 83-90
[6]Lian Q F,Li Y Z.Reducing subspace frame multiresolution analysis and frame wavelets.Commun Pure Appl Anal,2007,6:741-756
[7]Gu Q,Han D.Wavelet frames for(not necessarily reducing) affine subspaces.Appl Comput Harmon Anal, 2009,27:47-54
[8]Christensen O.An Introduction to Frames and Riesz Bases.Boston:Birkhauser,2003
[9]李云章,周凤英.L~2(R~d)中约化子空间上的仿射与伪仿射对偶小波标架.数学学报,2010,53:551-562
[10]de Boor C,DeVore R,Ron A.Approximation from shift-invariant subspaces of L~2(R~d).Trans Amer Math Soc,1994,341:787-806