Codes and Expansions (CodEx) Seminar

Alex Iosevich (University of Rochester)
On point configurations and frame theory

It has been known for some time that \(L^2(B_d)\) does not possess an orthogonal basis of exponentials, where \(B_d\) is the unit ball in \({\mathbb{R}}^d\), \(d \ge 2\). The proof, due to Iosevich, Katz, and Pedersen is based on the curvature properties of the boundary of the ball which is exploited using the Erdos Distance Problem from geometric combinatorics. It is reasonable to ask whether \(L^2(\sigma)\) has an orthogonal basis of exponentials, where \(\sigma\) is the natural measure on the sphere. It turns out that not only is the answer an emphatic no, but \(L^2(\sigma)\) does not even possess a frame of exponentials. We shall give a proof of this fact and discuss a number of related questions in frame theory.