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Invariant sets for QMF functions. (English) Zbl 1409.37049
Summary: A quadrature mirror filter (QMF) function can be considered as the transition function for a Markov process on the unit interval. The QMF functions that generate scaling functions for multiresolution analyses are then distinguished by properties of their invariant sets. By characterizing these sets, we answer in the affirmative a question raised by R. F. Gundy [Notices Am. Math. Soc. 57, No. 9, 1094–1104 (2010; Zbl 1208.60076)].
MSC:
37E05 Dynamical systems involving maps of the interval
37B10 Symbolic dynamics
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
60J35 Transition functions, generators and resolvents
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