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Three Problems on Exponential Bases
- Part of
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- Journal:
- Canadian Mathematical Bulletin / Volume 62 / Issue 1 / March 2019
- Published online by Cambridge University Press:
- 07 January 2019, pp. 55-70
- Print publication:
- March 2019
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- Article
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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 
$L^{2}(D)$.
