Hostname: page-component-6898cfcb55-w5hgh Total loading time: 0 Render date: 2026-11-11T05:01:57.165Z Has data issue: false hasContentIssue false
  • English
  • Français

Frames and Single Wavelets for Unitary Groups

Published online by Cambridge University Press:  20 November 2018

Eric Weber
Eric Weber*
Affiliation:
Department of Mathematics Texas, A&M University, College Station, TX 77843-3368, USA, email: weber@math.tamu.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We consider a unitary representation of a discrete countable abelian group on a separable Hilbert space which is associated to a cyclic generalized frame multiresolution analysis. We extend Robertson’s theorem to apply to frames generated by the action of the group. Within this setup we use Stone’s theorem and the theory of projection valued measures to analyze wandering frame collections. This yields a functional analytic method of constructing a wavelet from a generalized frame multiresolution analysis in terms of the frame scaling vectors. We then explicitly apply our results to the action of the integers given by translations on ${{L}^{2}}(\mathbb{R})$ .

Keywords

42C4043A2542C1546N99waveletmultiresolution analysisunitary group representationframe

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 3 , 01 June 2002, pp. 634 - 647
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Baggett, L., Carey, A., Moran, W. and Ohring, P., General Existence Theorems for Orthonormal Wavelets, An Abstract Approach. Publ. Res. Inst. Math. Sci. (1) 31 (1995), 95–111.Google Scholar
[2] Baggett, L., Courter, J. and Merrill, K., Construction ofWavelets from Generalized Conjugate Mirror Filters. Preprint, 2000.Google Scholar
[3] Baggett, L., Medina, H. and Merrill, K., Generalized Multiresolution Analyses, and a Construction Procedure for All Wavelet Sets in Rn . J. Fourier Anal. Appl. (6) 5 (1999), 563–573.Google Scholar
[4] Benedetto, J. and Li, S., The Theory of Multiresolution Analysis Frames and Applications to Filter Banks. Appl. Comput. Harmon. Anal. (4) 5 (1998), 389–427.Google Scholar
[5] Casazza, P., The Art of Frame Theory. Taiwanese Math, J.. (2) 4 (2000), 129–201.Google Scholar
[6] Courter, J., Construction of Dilation-d Orthonormal Wavelets. Ph.D. thesis, University of Colorado, Boulder, 1999.Google Scholar
[7] Daubechies, I., Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. (7) 41 (1988), 909–996.Google Scholar
[8] Halmos, P., Introduction to Hilbert Space and the Theory of Spectral Multiplicity. Chelsea, New York, 1957.Google Scholar
[9] Han, D. and Larson, D., Frames, Bases and Group Representations. Mem. Amer. Math. Soc. (697) 147(2000).Google Scholar
[10] Han, D., Larson, D., Papadakis, M. and Stavropoulos, T., Multiresolution Analyses of Abstract Hilbert Spaces and Wandering Subspaces. In: The Functional and Harmonic Analysis of Wavelets and Frames, Contemp. Math. 247 (1999), 259–284.Google Scholar
[11] Hernandez, E. and Weiss, G., An Introduction to Wavelets. CRC Press, Boca Raton, 1998.Google Scholar
[12] Lee, S., Tan, H. and Tang, W., Wavelet Bases for a Unitary Operator. Proc. Edinburgh Math. Soc. 38 (1995), 233–260.Google Scholar
[13] Mallat, S., Multiresolution Approximations and Wavelet Orthonormal Bases of L2(R). Trans. Amer. Math. Soc. (1) 315 (1989), 69–87.Google Scholar
[14] Papadakis, M., Generalized Frame MRAs of Abstract Hilbert Spaces. Preprint, 2000.Google Scholar
[15] Papadakis, M., On the dimension function of a wavelet. Proc. Amer. Math. Soc. (7) 128 (2000), 2043–2049.Google Scholar
[16] Robertson, J., On Wandering Subspaces for Unitary Operators. Proc. Amer. Math. Soc. 16 (1965), 233–236.Google Scholar
[17] Schaffer, S., Generalized Multiresolution Analyses and Applications of their Multiplicity Functions to Wavelets. Ph.D. thesis, University of Colorado, Boulder, 2000.Google Scholar
[18] Weber, E., On the translation invariance of wavelet subspaces. J. Fourier Anal. Appl. (5) 6 (2000), 551–558.Google Scholar