Wavelet sets without groups

TL;DR

This paper investigates the existence of wavelet sets for dilation sets that are not necessarily groups, providing a necessary and sufficient condition and illustrating the construction with examples.

Contribution

It introduces a new criterion for subspace wavelet sets for non-group dilation sets and demonstrates their construction through examples.

Findings

Established a necessary and sufficient condition for wavelet sets with non-group dilation sets

Provided explicit examples illustrating the construction of such wavelet sets

Extended the theory of wavelet sets beyond group-based dilation sets

Abstract

Most of the examples of wavelet sets are for dilation sets which are groups. We find a necessary and sufficient condition under which subspace wavelet sets exist for dilation sets of the form $A B$, which are not necessarily groups. We explain the construction by a few examples.