TY - JOUR
T1 - Examples of Riesz Bases of Exponentials for Multi-tiling Domains and Their Duals
AU - Frederick, Christina
AU - Yacoubou Djima, Karamatou
N1 - Publisher Copyright:
© The Author(s) 2023.
PY - 2024/3
Y1 - 2024/3
N2 - 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), where Ω 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.
AB - 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), where Ω 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.
KW - Paley Wiener spaces
KW - Riesz bases of exponentials
KW - Vandermonde matrices
UR - http://www.scopus.com/inward/record.url?scp=85195596254&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85195596254&partnerID=8YFLogxK
U2 - 10.1007/s44007-023-00078-7
DO - 10.1007/s44007-023-00078-7
M3 - Article
AN - SCOPUS:85195596254
SN - 2730-9657
VL - 3
SP - 108
EP - 123
JO - Matematica
JF - Matematica
IS - 1
ER -