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### Session S13 - Harmonic Analysis, Fractal Geometry, and Applications

Thursday, July 15, 16:35 ~ 17:05 UTC-3

## Completeness, exponentially completeness, and approximate orthogonality on the ball

### Azita Mayeli

Let $\Omega\subset \Bbb R^d$ be a bounded domain with positive Lebesgue measure, and assume that $\mathcal F\subset L^2(\Omega)$ is non-empty. We say $\mathcal F$ is {\it exponentially complete} if for all $\xi\in\Bbb R^d$, there is a function $f\in \mathcal F$ such that $$\langle f, e^{2\pi i x\cdot \xi} \rangle \neq 0 .$$ Let $B_d\subset \Bbb R^d$, $d>1$, denote the unit ball. It is known that $L^2(B_d)$ dose not admit any Parseval frame of exponentials. Motivated by this fact, we will investigate the exponentially completeness of exponential functions $\mathcal F=\mathcal E(A)\subset L^2(B_d)$, where $$\mathcal E(A):=\{f_a(x):=e^{2\pi i x\cdot a}: ~ a\in A\subset \Bbb R^d\},$$ and $A$ is countable. In particular, we show that for any set $A$ of size $\sharp A=2$, the set $\mathcal E(A)$ is exponentially incomplete in $L^2(B_d)$, $d\geq 2$, and also there are exponentially complete sets $\mathcal E(A)$ for any large size $A$ in $\Bbb R^d, ~d\geq 3$.\\
In the second half of the talk, we weaken the orthogonality condition and show that there is no set $A$ with positive and finite upper density such that the exponentials $\mathcal E(A)$ are mutually $\phi$-approximately orthogonal on the ball. More precisely, given a bounded domain $\Omega$, and a bounded measurable function $\phi:[0,\infty) \to [0, \infty)$ with $\phi(t)\to 0$ as $t\to \infty$, we say that $e^{2\pi i x\cdot a}$ and $e^{2\pi i x\cdot a'}$, $a\neq a'$, are {\it $\phi$-approximately orthogonal} if $$|\widehat{\chi_\Omega}(a-a')|\leq \phi(|a-a|).$$ We prove 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. \\