Methods of construction of exponential bases on planar domains

TL;DR

This paper surveys and refines methods for constructing exponential bases on planar domains, focusing on explicit frame bounds and applications to multi-tiling sets, enhancing the understanding of basis conditioning.

Contribution

It introduces refined techniques for constructing exponential bases with explicit control over frame bounds on planar domains, especially those that multi-tile the plane.

Findings

Constructed well-conditioned exponential bases on specific planar sets.

Provided explicit control over frame bounds in basis construction.

Applied techniques to multi-tiling planar domains.

Abstract

In this paper, we survey and refine several results -- some previously established in the literature -- that facilitate the construction of exponential bases on planar domains with explicit control over the associated frame bounds. We apply our techniques to construct well-conditioned exponential bases on certain planar sets that multi-tile the plane.