Frame-like Fourier expansions for finite Borel measures on $\mathbb{R}$

Abstract: 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 of pure type. In this paper, we prove the existence of a class of finite Borel measures $\mu$ on $[0,1)$ that are not of pure type that possess frame-like Fourier expansions for $L^{2}(\mu)$. We also show properties and classifications of certain measures possessing this type of Fourier expansion. Additionally, we establish a frame-like Fourier expansion for $L^{2}(\mu)$ where $\mu$ is a singular Borel probability measure on $\mathbb{R}$. Finally, we show measures $\mu$ on $[0,1)$ that possess these frame-like Fourier expansions for $L^{2}(\mu)$ have all $f\in L^{2}(\mu)$ as $L^{2}(\mu)$ limits of harmonic functions with frame-like coefficients. We also discuss when the inner products of these expansions coincide with model spaces and subspaces of harmonic functions on the disk.