Examples of Riesz Bases of Exponentials for Multi-tiling Domains and Their Duals

Abstract

A well-studied problem in sampling theory is to find an expansion of a function in terms of a Riesz basis of exponentials for L2(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(\\Omega )$$\\end{document}, where Ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega $$\\end{document} is a bounded, measurable set. For such a basis, we are guaranteed the existence of a unique biorthogonal dual basis that can be used to calculate the expansion coefficients. Much attention has been paid to the existence of Riesz bases of exponentials for various domains; however, the sampling and reconstruction problems in these cases are less understood. Recently, explicit formulas for the corresponding dual Riesz bases were introduced in Frederick and Okoudjou in [Appl Comput Harmon Anal 51:104–117, 2021; Frederick and Mayeli in J Fourier Anal Appl 27(5):1–21, 2021] for a class of multi-tiling domains. In this paper, we further this work by presenting explicit examples of a finite co-measurable union of intervals or multi-rectangles. In the higher-dimensional case, we also discuss how different sampling strategies lead to different Riesz bounds.