Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-28T15:59:22.068Z Has data issue: false hasContentIssue false
  • English
  • Français

Wavelet bases for a unitary operator

Published online by Cambridge University Press:  20 January 2009

S. L. Lee ,
H. H. Tan  and
W. S. Tang
S. L. Lee
Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore0511
H. H. Tan
Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore0511
W. S. Tang
Affiliation:
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore0511
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.

Let T be a unitary operator on a complex Hilbert space ℋ, and X, Y be finite subsets of ℋ. We give a necessary and sufficient condition for TZ(X): {Tnx: n ∈ Z, x ∈ X} to be a Riesz basis of its closed linear span 〈TZ(X)〉. If TZ(X) and TZ(Y) are Riesz bases, and 〈TZ(X)〉⊂〈TZ(Y)〉, then X is extendable to X′ such that TZ(X′) is a Riesz basis of TZ(Y) The proof provides an algorithm for the construction of Riesz bases for the orthogonal complement of 〈TZ(X)〉 in 〈TZ(Y)〉. In the case X consists of a single B-spline, the algorithm gives a natural and quick construction of the spline wavelets of Chui and Wang [2, 3]. Further, the duality principle of Chui and Wang in [3] and [4] is put in the general setting of biorthogonal Riesz bases in Hilbert space.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 38 , Issue 2 , June 1995 , pp. 233 - 260
Copyright
Copyright © Edinburgh Mathematical Society 1995

References

REFERENCES

1.Beylkin, G., Coifman, R. and Rokhlin, V., Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math. 44 (1991), 141–183.CrossRefGoogle Scholar
2.Chui, Charles K. and Wang, Jian-Zhong, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. 113 (1991), 785–793.CrossRefGoogle Scholar
3.Chui, Charles K. and Wang, Jian-Zhong, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc. 330 (1992), 903–915.CrossRefGoogle Scholar
4.Chui, Charles K. and Wang, Jian-Zhong, A general framework of compactly supported splines and wavelets, J. Approx. Theory 71 (1992), 263–304.CrossRefGoogle Scholar
5.Daubechies, I., Orthonormal bases of compactly supported wavelet, Comm. Pure Appl. Math. 41 (1988), 909–996.CrossRefGoogle Scholar
6.Daubechies, I., The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Infor. Theory 36 (1990), 961–1005.CrossRefGoogle Scholar
7.Edwards, R. E., Fourier series (Volume II, Holt, Rinehart and Winston, Inc., 1967).Google Scholar
8.Goodman, T. N. T., Lee, S. L. and Tang, W. S., Wavelets in wandering subspaces, Trans. Amer. Math. Soc. 338 (1993), 639–654.CrossRefGoogle Scholar
9.Halmos, P. R., A Hilbert space problem book, 2nd edition (Springer-Verlag, 1982).CrossRefGoogle Scholar
10.Lee, S. L., Tan, H. H. and Tang, W. S., Wavelet transformations and matrix compression (Proc. Internt. Conf. on Advances in Computational Mathematics, New Delhi, 1993).Google Scholar
11.Lemarié, P. G., Ondelettes a localisation exponentielles, J. Math. Pares Appl. 67 (1988), 227–236.Google Scholar
12.Mallat, S., Multiresolution approximations and wavelet orthonormal bases of L2(K), Trans. Amer. Math. Soc. 315 (1989), 69–87.Google Scholar
13.Meyer, Y., Ondelettes et functions splines (Seminaire Equations aux Derivees Partielles, Ecole Polytechnique, Paris, 1986).Google Scholar
14.Robertson, J. B., On wandering subspaces for unitary operators, Proc. Amer. Math. Soc. 16 (1965), 233–236.CrossRefGoogle Scholar
15.Schoenberg, I. J., Cardinal spline interpolation (CBMS-NSF Series in Appl. Math. 12, SIAM Publ., Philadelphia, 1973).CrossRefGoogle Scholar
16.Young, R. M., An introduction to nonharmonic Fourier series (Academic Press, 1980).Google Scholar