Wavelet frames and admissibility in higher dimensions

This paper is concerned with the relations between discrete and continuous wavelet transforms on k‐dimensional Euclidean space. We start with the construction of continuous wavelet transforms with the help of square‐integrable representations of certain semidirect products, thereby generalizing results of Bernier and Taylor. We then turn to frames of L2(Rk) and to the question, when the functions occurring in a given frame are admissible for a given continuous wavelet transform. For certain frames we give a characterization which generalizes a result of Daubechies to higher dimensions.

[1]  Syed Twareque Ali,et al.  Square integrability of group representations on homogeneous spaces. I. Reproducing triples and frames , 1991 .

[2]  Kendall E. Atkinson.  An introduction to numerical analysis , 1978 .

[3]  Ingrid Daubechies,et al.  Time-frequency localization operators: A geometric phase space approach , 1988, IEEE Trans. Inf. Theory.

[4]  J. Glimm,et al.  LOCALLY COMPACT TRANSFORMATION GROUPS( , 1961 .

[5]  Robert J. Zimmer,et al.  Ergodic Theory and Semisimple Groups , 1984 .

[6]  A. Carey,et al.  Square-integrable representations of non-unimodular groups , 1976, Bulletin of the Australian Mathematical Society.

[7]  A. Grossmann,et al.  Transforms associated to square integrable group representations. I. General results , 1985 .

[8]  Ingrid Daubechies,et al.  The wavelet transform, time-frequency localization and signal analysis , 1990, IEEE Trans. Inf. Theory.

[9]  Syed Twareque Ali,et al.  Square integrability of group representations on homogeneous spaces. II : Coherent and quasi-coherent states. The case of the Poincaré group , 1991 .

[10]  G. Mackey.  INDUCED REPRESENTATIONS OF LOCALLY COMPACT GROUPS I , 1952 .

[11]  David Bernier,et al.  Wavelets from square-integrable representations , 1996 .

[12]  Calvin C. Moore,et al.  On the regular representation of a nonunimodular locally compact group , 1976 .

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

[14]  G. Mackey.  Point realizations of transformation groups , 1962 .

[15]  Christopher Heil,et al.  Continuous and Discrete Wavelet Transforms , 1989, SIAM Rev..