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Optimal functions for a periodic uncertainty principle and multiresolution analysis*

Published online by Cambridge University Press:  20 January 2009

Jürgen Prestin
Affiliation:
Institute of Biomathematics and Biometry, GSF – National Research Center for Environment and Health 85764 Neuherberg, Germany, E-mail address: [email protected]
Ewald Quak
Affiliation:
Sintef Applied Mathematics, Postboks 124 Blindern N-0314 Oslo, Norway, E-mail address: [email protected]
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In this paper, it is shown that certain Theta functions are asymptotically optimal for the periodic time frequency uncertainty principle described by Breitenberger in [3]. These extremal functions give rise to a periodic multiresolution analysis where the corresponding wavelets also show similar localization properties.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1999

References

REFERENCES

1.Benedetto, J., Heil, C. and Walnut, D., Uncertainty principles for time-frequency operators, in Operator Theory: Advances and Applications, Vol. 58 (Birkhäuser Verlag, Basel, 1992), 125.Google Scholar
2.Berg, L., Asymptotische Darstellungen und Entwicklungen (Deutscher Verlag der Wissenschaften, Berlin, 1968).Google Scholar
3.Breitenberger, E., Uncertainty measures and uncertainty relations for angle observables, Found. Phys. 15 (1985), 353364.Google Scholar
4.Chui, C. K. and Wang, J. Z., A study of compactly supported scaling functions and wavelets, in Wavelets, Images and Surface Fitting (Laurent, P. J., Le Méhauté, A. and Schumaker, L. L. (eds.), AKPeters, Boston, 1994), 121140.Google Scholar
5.Chui, C. K. and Wang, J. Z., High-order orthonormal scaling functions and wavelets give poor time-frequency localization, J. Fourier Anal. Appl. 2 (1996), 415426.Google Scholar
6.Chui, C. K. and Wang, J. Z., A study of asymptotically optimal time-frequency localization by scaling functions and wavelets (CAT Report # 323, Texas A&M University, 1996).Google Scholar
7.Cohen, L., Time-frequency distributions – a review, Proc. IEEE 77 (1989), 941981.CrossRefGoogle Scholar
8.Daubechies, I., Ten lectures on wavelets (CBMS-NSF Series in Appl. Math., SIAM, Philadelphia, 1992).CrossRefGoogle Scholar
9.Koh, Y. W., Lee, S. L. and Tan, H. H., Periodic orthogonal splines and wavelets, Appl. Comput. Harmonic Anal. 2 (1995), 201218.Google Scholar
10.Narcowich, F. J. and Ward, J. D., Wavelets associated with periodic basis functions, Appl. Comput. Harmonic Anal. 3 (1996), 4056.CrossRefGoogle Scholar
11.Narcowich, F. J. and Ward, J. D., Non-stationary spherical wavelets for scattered data, in Approximation Theory VIII, Vol. 2: Wavelets and Multilevel Approximation (Chui, C. K. and Schumaker, L. L. (eds.), World Scientific, 1995), 301308.Google Scholar
12.Narcowich, F. J. and Ward, J. D., Non-stationary wavelets on the m-sphere for scattered data, Appl. Comput. Harmonic Anal. 3 (1996), 324336.CrossRefGoogle Scholar
13.Plonka, G. and Tasche, M., A unified approach to periodic wavelets, in Wavelets: theory, algorithms, and applications (Chui, C. K., Montefusco, L., and Puccio, L. (eds.), Academic Press, New York, 1994), 137151.Google Scholar
14.Prestin, J. and Quak, E., Time frequency localization of trigonometric Hermite operators, in Approximation Theory VIII, Vol. 2: Wavelets and Multilevel Approximation (Chui, C. K. and Schumaker, L. L. (eds.), World Scientific, 1995), 343350.Google Scholar
15.Rösler, M. and Voit, M., An uncertainty principle for ultraspherical expansions, J. Math. Anal. Appl. 209 (1997), 624634.Google Scholar
16.Selig, K., Trigonometric wavelets and the uncertainty principle, in Approximation Theory (Müller, M. W., Felten, M. and Mache, D. H. (eds.), Math. Research, Vol. 86, Akademie Verlag, Berlin, 1995), 293304.Google Scholar
17.Unser, M., Aldroubi, A. and Eden, M., On the asymptotic convergence of B-spline wavelets to Gabor functions, IEEE Trans. Inf. Theory 38 (1992), 864872.Google Scholar