Codes and Expansions (CodEx) Seminar

Azita Mayeli (City University of New York, the Graduate Center)Exponential completeness and $\phi$-approximate orthogonality on the unit ball

Assume that $\Omega\subset \mathbb{R}^d$ be a bounded domain with positive Lebesgue measure. Assume that $\mathcal F\subset L^2(\Omega)$ is a non-empty set. We say $\mathcal F$ is an exponentially complete system of functions, or is simply exponentially complete, if for all $\xi\in\mathbb{R}^d$, there is a function $f\in \mathcal F$ such that $\langle f, e^{2\pi i x\cdot \xi} \rangle \neq 0.$

Motivated by the problem of the non-existence of orthogonal bases of exponential functions on the unit ball in $\mathbb{R}^d$, $d>1$, in the first part of the talk we focus on the case where $\mathcal F=\mathcal E(A)$ is a system of exponential functions
$\mathcal E(A):=\{f_a(x):=e^{2\pi i x\cdot a}: ~ a\in A\subset \mathbb{R}^d\},$
$\Omega=B_d$ is the unit ball in $\mathbb{R}^d$,  $d>1$, and $A$ is a countable set, and we investigate the maximum (minimum) size of $A$ for which $\mathcal E(A)$ is exponentially complete (incomplete) with respect to the ambient space dimension.

Given a bounded domain $\Omega$, and a bounded measurable function $\phi:[0,\infty) \to [0, \infty)$ with $\phi(\xi)\to$ as $|\xi|\to \infty$, we say that $e^{2\pi i x\cdot a}$ and $e^{2\pi i x\cdot a'}$, $a\neq a'$, are $\phi$-approximately orthogonal if $|\widehat{\chi_\Omega}(a-a')|\leq \phi(|a-a|).$

In the second part of the talk, we show that if $\phi$ decays faster than $(1+t)^{-\frac{d+1}{2}}$ as $t\to \infty$, then the unit ball can not admit any $\phi$-approximate orthogonal basis of exponentials. This is a joint work with Alex Iosevich.