×

The time operator of wavelets. (English) Zbl 1115.82325

Summary: This paper establishes an interesting new and general connection between the wavelet theory of harmonic analysis and the Time operator theory of statistical physics. In particular, it will be shown that an arbitrary wavelet multiresolution analysis (MRA) defines a Time operator \(T\) whose age eigenspaces are the wavelet detail subspaces \(\mathcal W_n\). Extension of this result to the continuous parameter case induces a new notion of continuous wavelet multiresolution analysis. The Time operator \(T\) incorporates and exhibits in a natural way all five fundamental properties of a wavelet multiresolution analysis.

MSC:

82C05 Classical dynamic and nonequilibrium statistical mechanics (general)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
47B38 Linear operators on function spaces (general)
47N50 Applications of operator theory in the physical sciences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Daubechies, I., Ten Lectures on Wavelets (1992), SIAM Publications: SIAM Publications Philadelphia · Zbl 0776.42018
[2] Chui, C., An Introduction to Wavelets (1992), Academic Press: Academic Press Boston · Zbl 0925.42016
[3] Meyer, Y., Wavelets and Operators (1992), Cambridge Press: Cambridge Press Cambridge
[4] Heil, C.; Walnut, D., Continuous and discrete wavelet transforms, SIAM Review, 31, 628 (1989) · Zbl 0683.42031
[5] Strang, G., Wavelets and dilation equations: a brief introduction, SIAM Review, 31, 614 (1989) · Zbl 0683.42030
[6] Strang, G.; Wavelets, American Scientist, 82, 250 (1994)
[7] Briggs, W.; Henson, V.; Wavelets; multigrid, SIAM J. Sci. Comput., 14, 506 (1993)
[8] Strang, G.; Nguyen, T., Wavelets and Filter Banks (1996), Wesley-Cambridge Press: Wesley-Cambridge Press Boston · Zbl 1254.94002
[9] Y. Meyer, Ondelettes et fonctions splines, Seminaire EDP, Ecole Polytechnique, Paris, 1986; Y. Meyer, Ondelettes et fonctions splines, Seminaire EDP, Ecole Polytechnique, Paris, 1986
[10] S. Mallat, Multiresolution Approximations and Wavelets, Ph.D. Dissertation, University of Pennsylvania, 1988; S. Mallat, Multiresolution Approximations and Wavelets, Ph.D. Dissertation, University of Pennsylvania, 1988
[11] Chui, C., Multivariate Splines (1988), SIAM Publications: SIAM Publications Philadelphia · Zbl 0687.41018
[12] Burt, P.; Adelson, E., The Laplacian Pyramid as a compact image code, IEEE Trans. Comm., 31, 482 (1983)
[13] Jawerth, B.; Sweldens, W., An overview of wavelet based multiresolution analysis, SIAM Review, 36, 377 (1994) · Zbl 0803.42016
[14] A. Haar, Zur theorie der orthogonalen functionensystem, Math Annalen 69 (1910) 38; 71 (1911) 331; A. Haar, Zur theorie der orthogonalen functionensystem, Math Annalen 69 (1910) 38; 71 (1911) 331 · JFM 41.0469.03
[15] K. Gustafson, Partial Differential Equations, Wiley, New York, 1987; Dover, New Jersey, 1998; K. Gustafson, Partial Differential Equations, Wiley, New York, 1987; Dover, New Jersey, 1998 · Zbl 0623.35003
[16] Gustafson, K.; Misra, B., Canonical commutation relations of quantum mechanics and stochastic regularity, Letter in Math. Phys., 1, 119 (1976)
[17] Gustafson, K.; Goodrich, R., Kolmogorov systems and Haar systems, Colloq. Math. Soc. Janos Bolya, 49, 401 (1985)
[18] K. Gustafson, Lectures on Computational Fluid Dynamics, Mathematical Physics, and Linear Algebra, Kaigai, Tokyo, 1996; World Scientific, Singapore, 1997; K. Gustafson, Lectures on Computational Fluid Dynamics, Mathematical Physics, and Linear Algebra, Kaigai, Tokyo, 1996; World Scientific, Singapore, 1997
[19] Koopman, B., Hamiltonian systems and transformations in Hilbert spaces, Proc. Nat. Acad. Sci. USA, 17, 315 (1931) · JFM 57.1010.02
[20] Goodrich, R.; Gustafson, K.; Misra, B., On converse to Koopman’s Lemma, Physica A, 102, 379 (1980)
[21] Halmos, P., A Hilbert Space Problem Book (1982), Springer: Springer New York · Zbl 0202.12801
[22] Nagy, B. Sz.; Foias, C., Harmonic Analysis of Operators in Hilbert Space (1970), North Holland: North Holland Amsterdam · Zbl 0201.45003
[23] Mackey, G., Theory of Group Representations (1976), University of Chicago Press: University of Chicago Press Chicago
[24] W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik, in: Handbuch der Physik, S. Flugge (Eds.), Springer, Berlin, 1958; W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik, in: Handbuch der Physik, S. Flugge (Eds.), Springer, Berlin, 1958 · Zbl 0007.13504
[25] Misra, B.; Prigogine, I.; Courbage, M., From deterministic dynamics to probabilistic descriptions, Physica, 38A, 1 (1979) · Zbl 0411.60100
[26] Antoniou, I.; Misra, B., Relativistic internal Time operator, Intern. J. of Theor. Phys., 31, 119 (1992) · Zbl 0812.46069
[27] Cornfeld, I.; Fomin, S.; Sinai, Ya, Ergodic Theory (1982), Springer: Springer Berlin
[28] Putnam, C., Commutation Properties of Hilbert Space Operators and Related Topics (1967), Springer: Springer Berlin · Zbl 0149.35104
[29] Lax, P.; Phillips, R. S., Scattering Theory (1967), Academic: Academic New York · Zbl 0214.12002
[30] I. Antoniou, K. Gustafson, Wavelets and Stochastic Processes, to appear; I. Antoniou, K. Gustafson, Wavelets and Stochastic Processes, to appear · Zbl 0933.60061
[31] M. Stone, Linear Transformations in Hilbert Space, Amer. Math. Soc., Providence, R.I., 1932; M. Stone, Linear Transformations in Hilbert Space, Amer. Math. Soc., Providence, R.I., 1932 · JFM 58.0420.02
[32] Marti, J., Introduction to the Theory of Bases (1969), Springer: Springer New York · Zbl 0191.41301
[33] Ali, S.; Antoine, J. P.; Gazeau, J. P.; Mueller, U., Coherent states and their generalizations: a mathematical overview, Reviews in Mathematical Physics, 7, 1013 (1995) · Zbl 0837.43014
[34] Flornes, K.; Grossmann, A.; Holschneider, M.; Torrésani, B., Wavelets on discrete fields, Applied Comput. Harm. Analysis, 1, 137 (1994) · Zbl 0798.42021
[35] Duval-Destin, M.; Muschietti, M.; Torresani, B., Continuous wavelet decompositions, multiresolutions, and contrast analysis, SIAM J. Math Anal., 24, 739 (1993) · Zbl 0770.41023
[36] Stark, H., Continuous wavelet transform and continuous multiscale analysis, J. Math. Anal. Applic., 169, 179 (1992) · Zbl 0766.42015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.