Interpolation maps and congruence domains for wavelet sets

TL;DR

This paper investigates properties of interpolation maps between wavelet sets, proving conditions for when pairs are interpolation pairs and providing a counterexample that challenges previous assumptions, with implications for wavelet theory.

Contribution

It proves that union-preserving interpolation maps imply interpolation pairs and constructs a counterexample showing congruence domains do not guarantee an interpolation pair.

Findings

Union-preserving interpolation maps imply interpolation pairs.

Counterexample where congruence domains are equal but the pair is not an interpolation pair.

Applications and technical lemmas related to wavelet sets.

Abstract

It is proven that if an interpolation map between two wavelet sets preserves the union of the sets, then the pair must be an interpolation pair. We also construct an example of a pair of wavelet sets for which the congruence domains of the associated interpolation map and its inverse are equal, and yet the pair is not an interpolation pair. The first result solves affirmatively a problem that the second author had posed several years ago, and the second result solves an intriguing problem of D. Han. The key to this counterexample is a special technical lemma on constructing wavelet sets. Several other applications of this result are also given. In addition, some problems are posed. We also take the opportunity to give some general exposition on wavelet sets and operator-theoretic interpolation of wavelets.