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.