A Fourier Frame for the Middle-Third Cantor Measure

Abstract: In this paper we show that if $\mu$ is any locally and uniformly $\alpha$-dimensional measure supported on a $\alpha$-quasi-regular set $E$, then $L<sup>2(\mu)$ admits a frame of exponentials. In particular, for the uniform middle third Cantor measure, $\mu_C,$ our result shows that there exists a countable set $\Lambda$ such that $${e^{2\pi i t \lambda}}_{\lambda \in \Lambda}$$ is a frame for $L<sup>2(\mu_C)$ (i.e. the measure $\mu_C$ admits a generalized spectrum), answering an old outstanding question about the existence of a frame of exponentials for the space $L<sup>2(\mu_C)$.