Methods of construction of exponential bases on planar domains

TL;DR

The paper addresses the problem of constructing exponential bases on planar domains with explicit frame bounds. It develops a constructive framework that combines: (i) tiling-based domain decomposition into unions of axis-aligned rectangles, (ii) lifting 1D exponential bases to 2D using a compatible lattice structure, and (iii) Vandermonde-based spectral estimates to bound the resulting frame constants. The authors provide explicit examples, including an octagon, and establish tools to control the frame bounds via matrix singular values and clustering. The results yield well-conditioned exponential bases on domains that tile the plane, enabling stable representations and reconstructions in applications requiring planar harmonic analysis.

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.

Figures (5)

• Figure 1: Octagon
• Figure 2: A rhombus with diagonals 2a and 2b
• Figure 3: The domain $D=R_1\cup R_2$
• Figure 4: Approximation of $D$ by $D_N$
• Figure 5: Union of cubes generated by $A_{N,2}$

Theorems & Definitions (22)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• Theorem 2.6
• Lemma 2.7
• proof : Proof of Theorem \ref{['T-Segm-rect']}