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W-Markov measures, transfer operators, wavelets and multiresolutions
Book chapter   Open access

W-Markov measures, transfer operators, wavelets and multiresolutions

Daniel Alpay, Palle E T Jorgensen and Izchak Lewkowicz
Frames and Harmonic Analysis: AMS special sessions on frames, wavelets, and Gabor systems and frames, harmonic analysis, and operator theory, April 16–17, 2016, North Dakota State University, Fargo, North Dakota, pp.293-343
Contemporary mathematics , 706, American mathematical Society
2018
DOI: 10.1090/conm/706/14219
url
https://doi.org/10.1090/conm/706/14219View
Published (Version of record) Open Access

Abstract

In a general setting we solve the following inverse problem: Given a positive operator R, acting on measurable functions on a fixed measure space (X, BX), we construct an associated Markov chain. Specifically, starting with a choice of R (the transfer operator), and a probability measure μ0 on (X,BX), we then build an associated Markov chain T0, T1, T2,..., with these random variables (r.v) realized in a suitable probability space (Ω, F, P), and each r.v. taking values in X, and with T0 having the probability μ0 as law. We further show how spectral data for R, e.g., the presence of R-harmonic functions, propagate to the Markov chain. Conversely, in a general setting, we show that every Markov chain is determined by its transfer operator. In a range of examples we put this correspondence into practical terms: (i) iterated function systems (IFS), (ii) wavelet multiresolution constructions, and (iii) IFSs with random “control.” Our setting for IFSs is general as well: a fixed measure space (X,BX) and a system of mappings τi, each acting in (X,BX), and each assigned a probability, say pi which may or may not be a function of x. For standard IFSs, the pi’s are constant, but for wavelet constructions, we have functions pi(x) reflecting the multi-band filters which make up the wavelet algorithm at hand. The sets τi(X) partition X, but they may have overlap, or not. For IFSs with random control, we show how the setting of transfer operators translates into explicit Markov moves: Starting with a point x ∈ X, the Markov move to the next point is in two steps, combined yielding the move from T0 = x to T1 = y, and more generally from Tn to Tn+1. The initial point x will first move to one of the sets τi(X) with probability pi, and once there, it will “choose” a definite position y (within τi(X)), now governed by a fixed law (a given probability distribution). For Markov chains, the law is the same in each move from Tn to Tn+1.

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