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Wavelets and stochastic processes. (English) Zbl 0933.60061

Summary: Wavelets are known to have intimate connections to several other parts of mathematics, notably phase-space analysis of signal processing, reproducing kernel Hilbert spaces, coherent states in quantum mechanics, spline approximation theory, windowed Fourier transforms, and filter banks. Here, we establish and survey a new connection, namely to stochastic processes. Key to this link are the Kolmogorov systems of ergodic theory.

MSC:

60G99 Stochastic processes
60G35 Signal detection and filtering (aspects of stochastic processes)
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[1] I. Antoniou, K. Gustafson, The time operator of wavelets, Chaos, Solitons, Fractals (to appear) · Zbl 1115.82325
[2] Antoniou, I.; Misra, B.: Relativistic internal time operator. Intern. J. Theor. phys. 31, 119-136 (1992) · Zbl 0812.46069
[3] Briggs, W.; Henson, V.: Wavelets and multigrid. SIAM J. Sci. comput. 14, 506-510 (1993) · Zbl 0773.65079
[4] Burt, P.; Adelson, E.: The Laplacian pyramid as a compact image code. IEEE trans. Comm. 31, 482-540 (1983)
[5] C. Chiann, P. Morettin, A Wavelet Analysis for Stationary Processes, University of Sao Paulo, Brazil, 1996, preprint · Zbl 0922.62094
[6] C. Chui, Multivariate Splines, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1988
[7] C. Chui, An Introduction to Wavelets, Academic Press, Boston, MA, 1992 · Zbl 0925.42016
[8] Cooley, J.; Tukey, J.: An algorithm for the machine calculation of complex Fourier series. Math. comp. 19, 297-301 (1965) · Zbl 0127.09002
[9] I. Cornfeld, S. Fomin, Ya. Sinai, Ergodic Theory, Springer, Berlin, 1982
[10] H. Cramer, M. Leadbetter, Stationary and Related Stochastic Processes, Wiley, New York, 1967 · Zbl 0162.21102
[11] Daubechies, I.: Orthonormal bases of compactly supported wavelets. Comm. pure appl. Math. 41, 909-996 (1988) · Zbl 0644.42026
[12] I. Daubechies, Orthonormal bases of wavelets with finite support-connection with discrete filters, in J. Combes, A. Grossmann, P. Tchamitchian (Eds.), Wavelets, Springer, Berlin, 1989, pp. 38–66 · Zbl 0850.42013
[13] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992 · Zbl 0776.42018
[14] D. Donoho, Nonlinear Wavelet Methods for Recovery of Signals, Densities, and Spectra from Indirect and Noisy Data, Proc. Symp. Appl. Math, 47; Amer. Math. Soc. Providence, RI, 1993, pp. 173–205 · Zbl 0786.62094
[15] J. Doob, Stochastic Processes, Wiley, New York, 1953 · Zbl 0053.26802
[16] Gabor, D.: Theory of communication. J. ieee 93, 429-457 (1946)
[17] R. Gopinath, C. Burris, Wavelet transforms and filter banks, in Wavelets: A Tutorial in Theory and Applications, C. Chui (Ed.), Academic Press, Boston, MA, 1992, pp. 603–654 · Zbl 0776.42022
[18] K. Gustafson, Lectures on Computational Fluid Dynamics, Mathematical Physics and Linear Algebra, Kaigai, Tokyo, Japan, 1996; World Scientific, Singapore, 1997
[19] K. Gustafson, Operator trigonometry of wavelet frames, in: J. Wang, M. Allen, B. Chen, T. Mathew (Eds.), Iterative Methods in Scientific Computation; IMACS Series in Computational and Applied Mathematics 4, New Brunswick, NJ, 1998, pp. 161–166
[20] Gustafson, K.; Goodrich, R.: Kolmogorov systems and Haar systems. Colloq. math. Soc. janos bolyai 49, 401-416 (1985)
[21] Gustafson, K.; Misra, B.: Canonical commutation relations of quantum mechanics and stochastic regularity. Lett. math. Phys. 1, 275-280 (1976) · Zbl 0345.60022
[22] A. Haar, Zur Theorie der Orthogonalen Functionensystem I, II, Math. Annalen 69 (1910), pp. 38–53; 71 (1911), pp. 331–371
[23] P. Halmos, A Hilbert Space Problem Book, Springer, New York, 1982
[24] Heil, C.; Walnut, D.: Continuous and discrete wavelet transforms. SIAM review 31, 628-666 (1989) · Zbl 0683.42031
[25] T. Hida, Stationary Stochastic Processes, Princeton University Press, Princeton, NJ, 1970 · Zbl 0214.16401
[26] Jawerth, B.; Sweldens, W.: An overview of wavelet based multiresolution analyses. SIAM review 36, 377-412 (1994) · Zbl 0803.42016
[27] B. Koopman, Hamiltonian systems and transformations in Hilbert spaces, Proc. Nat. Acad. Sci USA, 17 (1931) 377–412 · Zbl 0002.05701
[28] Koopman, B.; Von Neumann, J.: Dynamical systems of continuous spectra. Proc. nat. Acad. sci. USA 18, 255-266 (1932) · Zbl 0006.22702
[29] P. Kopp, Martingales and Stochastic Integrals, Cambridge University Press, Cambridge, UK, 1984 · Zbl 0537.60047
[30] U. Krengel, Ergodic Theorems, De Gruyter, Berlin, 1985
[31] P. Lax, R. Phillips, Scattering Theory, Academic Press, New York, 1967
[32] G. Mackey, theory of Group Representations, University of Chicago Press, Chicago, 1976 · Zbl 0344.22002
[33] Mallat, S.: Multiresolution approximations and wavelet orthonormal bases of \(L2(R)\). Trans. am. Math. soc. 315, 69-87 (1989) · Zbl 0686.42018
[34] Y. Meyer, Ondelettes et fonctions spline, Seminar EDP, Ecole Polytechnique, Paris, 1986
[35] Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1992 · Zbl 0776.42019
[36] G. Nason, Wavelet regression by cross-validation, TR 447, Dept. of Statistics, Stanford University, 1994
[37] Neveu, J.: Relations entre la theorie des martingales et al theorie ergodic. Ann. inst. Fourier Grenoble 15, 31-42 (1965) · Zbl 0202.47202
[38] W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik, in: S. Flugge (Ed.), Handbuch der Physik, Springer, Berlin, 1958, pp. 1–168 · Zbl 0007.13504
[39] H. Primas, Chemistry, Quantum Mechanics and Reductionism, Spring Lecture Notes in Chemistry, 24, Berlin, 1981
[40] C. Putnam, Commutation Properties of Hilbert Space Operators and Related Topics, Springer, Berlin, 1967 · Zbl 0149.35104
[41] M. Rao, Abstract martingales and ergodic theory, in Multivariate Analysis III, Academic Press, New York, 1972, pp. 45–60
[42] E. Robinson, Predictive decomposition of time series with application to seismic exploration, MIT GAG Rept. No. 7 (1954); reprinted in, G. Webster (Ed.), Deconvolution, Society of Exploration Geophysicists Tulsa, OK, 1978, pp. 52–118
[43] Y. Rozanov, Innovation Processes, Wiley, New York, 1977 · Zbl 0359.60005
[44] Smith, M.; Barnwell, T.: Exact reconstruction techniques for tree-structured subband coders. IEEE trans. Acoustics 34, 434-441 (1986)
[45] B. Sz. Nagy, C. Foias, Harmonic Analysis of Operators in Hilbert Space, North Holland, Amsterdam, 1970
[46] Strang, G.: Wavelets and dilation equations: a brief introduction. SIAM rev. 31, 614-627 (1989) · Zbl 0683.42030
[47] Strang, G.: Wavelets. Am. scientist 82, 250-255 (1994)
[48] G. Strang, T. Nguyen, Wavelets and Filter Banks, Wellesley–Cambridge Press, Boston, MA, 1996
[49] P. Walters, Ergodic Theory, Springer Lecture Notes in Math. 458, 1975, Berlin
[50] G. Webster (Ed.), Deconvolution, Society of Exploration Geophysics, Tulsa, OK, 1978
[51] Ali, S.; Antoine, J. P.; Gazeau, J. P.; Mueller, U.: Coherent states and their generalizations: a mathematical overview. Rev. math. Phys. 7, 1013-1104 (1995) · Zbl 0837.43014
[52] Duval-Destin, M.; Muschietti, A.; Torresani, B.: Continuous wavelet decompositions, multiresolutions, and contrast analysis. SIAM J. Math. anal. 24, 739 (1993) · Zbl 0770.41023
[53] Stark, H.: Continuous wavelet transform and continuous multiscale analysis. J. math. Anal. appl. 169, 179-196 (1992) · Zbl 0766.42015
[54] J. Pham, Unilateral shifts in wavelet theory and algorithm, in Abstracts, 1st International Conference on Semigroups of Operators: Theory and Applications, Newport Beach, CA, 14–18 December 1998, p. 34; also see J. Pham, dissertation, electrical engineering, UCLA, 1999 (to appear)
[55] Goodman, T. N. T.; Lee, S. L.; Tang, W. S.: Wavelets in wandering subspaces. Trans. am. Math. soc. 228, 639-654 (1993) · Zbl 0777.41011
[56] X. Dai, D.R. Larson, Wandering Vectors for Unitary Systems and Orthogonal Wavelets, American Math. Soc. Memoirs 640, Providence, RI, 1998 · Zbl 0990.42022
[57] K. Gustafson, Wavelets as stochastic processes, in M. Kobayashi, S. Sakakibara, M. Yamada (Eds.), Proc. Workshop on Wavelets and Wavelet-based Technologies, Tokyo, Japan, 29–30 October 1998, IBM Japan/University of Tokyo, 1998, pp. 40–43
[58] T. Irino, R. Patterson, A time-domain, level-dependent auditory filter: the gamma chirp, in M. Kobayashi, S. Sakakibara, M. Yamada (Eds.), Proc. Workshop on Wavelets and Wavelet-based Technologies, Tokyo, Japan, 29–30 October 1998, IBM Japan/University of Tokyo, 1998, 44–59; see also J. Acoustical Soc. Am. 101 (1997)
[59] M. Kobayashi, M. Sakamoto, T. Saito, Y. Hashimoto, M. Nishimura, K. Suzuki, Wavelet Analysis for a Text-to-Speech (TTS) System, in: M. Koyayashi (Ed.), Wavelets and Their Applications: Case Studies, SIAM Publications, Philadelphia, PA, 1998, pp. 75–100 · Zbl 1052.68714
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