TL;DR
This paper investigates the invariant sets of QMF functions, which are crucial for generating scaling functions in multiresolution analysis, providing a characterization that answers a previously posed mathematical question.
Contribution
It offers a new characterization of invariant sets for QMF functions, advancing understanding of their role in multiresolution analysis.
Findings
Characterization of invariant sets for QMF functions
Answer to Gundy's question on invariant sets
Enhanced understanding of QMF functions in wavelet theory
Abstract
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 Gundy (Notices Amer. Math. Soc. 57, 1094-1104, 2010).
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