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\(W\)-Markov measures, transfer operators, wavelets and multiresolutions. (English) Zbl 1398.37020

Kim, Yeonhyang (ed.) et al., Frames and harmonic analysis. AMS special session on frames, wavelets and Gabor systems and special session on frames, harmonic analysis, and operator theory, North Dakota State University, Fargo, ND, USA, April 16–17, 2016. Proceedings. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-3619-3/pbk; 978-1-4704-4723-6/ebook). Contemporary Mathematics 706, 293-343 (2018).
Summary: In a general setting we solve the following inverse problem: Given a positive operator \(R\), acting on measurable functions on a fixed measure space \((X,\mathcal B_X)\), we construct an associated Markov chain. Specifically, starting with a choice of \(R\) (the transfer operator), and a probability measure \(\mu _0\) on \((X, \mathcal B_X)\), we then build an associated Markov chain \(T_0, T_1, T_2,\dots \), with these random variables (r.v) realized in a suitable probability space \((\Omega ,\mathcal F, \mathbb P)\), and each r.v. taking values in \(X\), and with \(T_0\) having the probability \(\mu _0\) 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 \((X, \mathcal B_X)\) and a system of mappings \(\tau _i\), each acting in \((X, \mathcal B_X)\), and each assigned a probability, say \(p_i\) which may or may not be a function of \(x\). For standard IFSs, the \(p_i\)’s are constant, but for wavelet constructions, we have functions \(p_i(x)\) reflecting the multi-band filters which make up the wavelet algorithm at hand. The sets \(\tau _i(X)\) 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 \(x\in X\), the Markov move to the next point is in two steps, combined yielding the move from \(T_0 = x\) to \(T_1 = y\), and more generally from \(T_n\) to \(T_{n+1}\). The initial point \(x\) will first move to one of the sets \(\tau _i(X)\) with probability \(p_i\), and once there, it will “choose” a definite position \(y\) (within \(\tau _i(X)\)), now governed by a fixed law (a given probability distribution). For Markov chains, the law is the same in each move from \(T_n\) to \(T_{n+1}\).
For the entire collection see [Zbl 1390.42001].

MSC:

37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
46L55 Noncommutative dynamical systems
47B65 Positive linear operators and order-bounded operators
60J05 Discrete-time Markov processes on general state spaces
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
65T60 Numerical methods for wavelets
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