Wavelet constructions in non-linear dynamics
Authors:
Dorin Ervin Dutkay and Palle E.T. Jorgensen
Journal:
Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 21-33
MSC (2000):
Primary 60G18; Secondary 42C40, 46G15, 42A65, 28A50, 30D05, 47D07, 37F20
DOI:
https://doi.org/10.1090/S1079-6762-05-00143-5
Published electronically:
March 7, 2005
MathSciNet review:
2122446
Full-text PDF Free Access
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Abstract: We construct certain Hilbert spaces associated with a class of non-linear dynamical systems $X$. These are systems which arise from a generalized self-similarity and an iterated substitution. We show that when a weight function $W$ on $X$ is given, then we may construct associated Hilbert spaces $H(W)$ of $L^2$-martingales which have wavelet bases.
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- V. Baladi, Positive Transfer Operators and Decay of Correlations, World Scientific, River Edge, NJ, 2000.
- S. Bildea, D. Dutkay, G. Picioroaga, MRA Super-wavelets, preprint 2004.
- M. Bramson, S. Kalikow, Nonuniqueness in $g$-functions, Israel J. Math. 84 (1993), 153â160.
- H. Brolin, Invariant sets under iteration of rational functions, Ark. Mat. 6 (1965), 103â144.
- I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, SIAM, Philadelphia, 1992.
- J.L. Doob, The Brownian movement and stochastic equations, Ann. of Math. (2) 43 (1942), 351â369.
- ---, What is a martingale? Amer. Math. Monthly 78 (1971), 451â463.
- ---, Classical Potential Theory and Its Probabilistic Counterpart, Grundlehren 262, Springer-Verlag, New York, 1984.
- D.E. Dutkay, P.E.T. Jorgensen, Wavelets on fractals, preprint, University of Iowa, 2003, http://arXiv.org/abs/math.CA/0305443 , to appear in Rev. Mat. Iberoamericana.
- ---, Martingales, endomorphisms, and covariant systems of operators in Hilbert space, preprint, University of Iowa, 2004, http://arXiv.org/abs/math.CA/0407330 .
- ---, Operators, martingales, and measures on projective limit spaces, preprint, University of Iowa, 2004, http://arxiv.org/abs/math.CA/0407517 .
- ---, Disintegration of projective measures, preprint, University of Iowa, 2004, http://arxiv.org/abs/math.CA/0408151 .
- R.F. Gundy, Low-pass filters, martingales, and multiresolution analyses, Appl. Comput. Harmon. Anal. 9 (2000), 204â219.
- ---, Wavelets and probability, preprint, Rutgers University, material presented during the authorâs lecture at the workshop âWavelets and Applicationsâ, Barcelona, Spain, July 1â6, 2002, http://www.imub.ub.es/wavelets/Gundy.pdf .
- ---, Martingale theory and pointwise convergence of certain orthogonal series, Trans. Amer. Math. Soc. 124 (1966), 228â248.
- ---, Two remarks concerning wavelets: Cohenâs criterion for low-pass filters and Meyerâs theorem on linear independence, The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, 1999) (L.W. Baggett, D.R. Larson, eds.), Contemp. Math., vol. 247, American Mathematical Society, Providence, 1999, pp. 249â258.
- E. Nelson, Topics in Dynamics I: Flows, Mathematical Notes, Princeton University Press, Princeton, NJ, 1969.
- J. Neveu, Discrete-Parameter Martingales, North-Holland, Amsterdam-Oxford, 1975.
- D. Ruelle, The thermodynamic formalism for expanding maps, Comm. Math. Phys. 125 (1989), 239â262.
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Additional Information
Dorin Ervin Dutkay
Affiliation:
Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419
MR Author ID:
608228
Email:
ddutkay@math.rutgers.edu
Palle E.T. Jorgensen
Affiliation:
Department of Mathematics, The University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419
MR Author ID:
95800
ORCID:
0000-0003-2681-5753
Email:
jorgen@math.uiowa.edu
Keywords:
Measures,
projective limits,
transfer operator,
martingale,
fixed point,
multiresolution,
Julia set,
subshift,
wavelet
Received by editor(s):
October 28, 2004
Published electronically:
March 7, 2005
Communicated by:
Boris Hasselblatt
Article copyright:
© Copyright 2005
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
