Codes and Expansions (CodEx) Seminar

Chad Berner (Iowa State University):
Frame-like Fourier expansions for finite Borel measures

Fourier series approximation in \(L^{2}([0,1))\) is effective because of the orthogonality of the exponential functions. Additionally, Fourier approximation with respect to other measures also proves useful in areas of probability and signal recovery. Sometimes other measures even possess a frame of exponential functions, which allows for Fourier expansions with many desirable properties. Furthermore, it is known that if a finite Borel measure \(\mu \) on \([0,1)\) possesses a frame of exponential functions for \(L^{2}(\mu )\), then \(\mu \) is absolutely continuous or singular. In this talk, we discuss the existence of a class of finite Borel measures \(\mu \) on \([0,1)\) that are not absolutely continuous or singular that possess Fourier frame-like expansions, which preserve some of the desirable properties of Fourier frame expansions. Finally, we discuss the classification of measures that possess Fourier frame-like expansions arising from operator orbits.