Analysis Seminar | Alex Iosevich (University of Rochester)

Title: Exact signal recovery, restriction phenomenon, and applications

Abstract: Let $f: {\mathbb Z}_N^d \to {\mathbb C}$ be a signal and suppose that the frequencies ${\{\widehat{f}(m)\}}_{m \in S}$ are missing, where
 $$\widehat{f}(m)=N^{-\frac{d}{2}} \sum_{x \in {\mathbb Z}_N^d} e^{-\frac{2 \pi i x \cdot m}{N}} f(x).$$

Under what reasonable assumption can we recover the original signal? We are going to discuss the connections between this problem and the classical restriction phenomenon in harmonic analysis. Bourgain's celebrated $\Lambda(q)$ theorem and its variants by Talagrand and others play an important role in our investigations. 

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