Alex Iosevich: Signal recovery, restriction theory, and applications IV

Let $f: {\mathbb Z}_N^d \to {\mathbb C}$ be a signal and suppose that it is transmitted via its Fourier transform
$\widehat{f}(m)=N^{-d} \sum_{x \in {\mathbb Z}_N^d} \chi(-x \cdot m) f(x)$,
where the frequencies
${\{\widehat{f}(m)\}}_{m \in S}$
are missing, for some $S \subset {\mathbb Z}_N^d$.

The question we ask is, can the original signal $f$ be recovered exactly? Matolcsi and Szuchs, and Donoho and Stark (independently) proved that if $f$ is supported in $E \subset {\mathbb Z}_N^d$ with
$|E||S| less than \frac{N^d}{2}$,
then $f$ can be recovered exactly. We are going to see that if $S$ satisfies a non-trivial restriction estimate, the recovery condition can be significantly improved. We are also going to see that multi-linear restriction theory can be used to improve the recovery condition for multiple transmissions.

Continuous aspects of this problem will also be discussed with the restriction conjecture described as a signal recovery mechanism. This is joint work with Azita Mayeli (CUNY).