Geometric aspects of frame representations of abelian groups

hhWe consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.

[1]  Igor Kluvánek,et al.  Sampling Theorem in Abstract Harmonic Analysis , 1965 .

[2]  B. Jawerth,et al.  A discrete transform and decompositions of distribution spaces , 1990 .

[3]  Charles K. Chui,et al.  oversampling preserves any tight affine frame for odd , 1994 .

[4]  Charles K. Chui,et al.  N oversampling preserves any tight a ne frame for odd n , 1994 .

[5]  A. Faridani.  A generalized sampling theorem for locally compact abelian groups , 1994 .

[6]  Zuowei Shen.  Affine systems in L 2 ( IR d ) : the analysis of the analysis operator , 1995 .

[7]  G. Folland.  A course in abstract harmonic analysis , 1995 .

[8]  David R. Larson,et al.  Wavelet sets in ℝn , 1997 .

[9]  A. Ron,et al.  Affine Systems inL2(Rd): The Analysis of the Analysis Operator , 1997 .

[10]  J. Benedetto,et al.  The Theory of Multiresolution Analysis Frames and Applications to Filter Banks , 1998 .

[11]  Akram Aldroubi,et al.  Wavelets, Multiwavelets, and Their Applications , 1998 .

[12]  Herbert A. Medina,et al.  GENERALIZED MULTIRESOLUTION ANALYSES, AND A CONSTRUCTION PROCEDURE FOR ALL WAVELET SETS IN R , 1998 .

[13]  R. L. Stens,et al.  Sampling theory in Fourier and signal analysis : advanced topics , 1999 .

[14]  Herbert A. Medina,et al.  Generalized multi-resolution analyses and a construction procedure for all wavelet sets in ℝn , 1999 .

[15]  Compactly Supported Wavelets and Representations of the Cuntz Relations , 1999, math/9912129.

[16]  An Abstract Interpretation of the Wavelet Dimension Function Using Group Representations , 2000 .

[17]  Deguang Han,et al.  Frames, bases, and group representations , 2000 .

[18]  P. Jorgensen.  MINIMALITY OF THE DATA IN WAVELET FILTERS , 2000, math/0004098.

[19]  John J. Benedetto,et al.  Wavelet Frames: Multiresolution Analysis and Extension Principles , 2001 .

[20]  Hamid Behmard,et al.  Sampling of Bandlimited Functions on Unions of Shifted Lattices , 2002 .

[21]  Chris Chatfteld,et al.  Wavelet Transforms and Time-Frequency Signal Analysis , 2002, Technometrics.

[22]  C. Chui,et al.  Characterization of General Tight Wavelet Frames with Matrix Dilations and Tightness Preserving Oversampling , 2002 .

[23]  E. Weber.  Frames and Single Wavelets for Unitary Groups , 2002, Canadian Journal of Mathematics.

[24]  Hans G. Feichtinger,et al.  Recovery of Band-Limited Functions on Locally Compact Abelian Groups from Irregular Samples , 2003 .