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Three Problems on Exponential Bases

Part of: Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  07 January 2019

Laura De Carli ,
Alberto Mizrahi  and
Alexander Tepper
Laura De Carli
Affiliation:
Department of Mathematics, Florida International University, Miami, FL 33199, USA Email: decarlil@fiu.eduamizr007@fiu.eduatepp001@fiu.edu
Alberto Mizrahi
Affiliation:
Department of Mathematics, Florida International University, Miami, FL 33199, USA Email: decarlil@fiu.eduamizr007@fiu.eduatepp001@fiu.edu
Alexander Tepper
Affiliation:
Department of Mathematics, Florida International University, Miami, FL 33199, USA Email: decarlil@fiu.eduamizr007@fiu.eduatepp001@fiu.edu
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Abstract

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We consider three special and significant cases of the following problem. Let $D\subset \mathbb{R}^{d}$ be a (possibly unbounded) set of finite Lebesgue measure. Let $E(\mathbb{Z}^{d})=\{e^{2\unicode[STIX]{x1D70B}ix\cdot n}\}\text{}_{n\in \mathbb{Z}^{d}}$ be the standard exponential basis on the unit cube of $\mathbb{R}^{d}$. Find conditions on $D$ for which $E(\mathbb{Z}^{d})$ is a frame, a Riesz sequence, or a Riesz basis for $L^{2}(D)$.

Keywords

exponential basisframeRiesz sequencelattice

MSC classification

Primary: 42C15: General harmonic expansions, frames
Secondary: 42C30: Completeness of sets of functions
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 1 , March 2019 , pp. 55 - 70
Copyright
© Canadian Mathematical Society 2018 

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