A Fourier Frame for the Middle-Third Cantor Measure

TL;DR

This paper proves that certain fractal measures, including the middle-third Cantor measure, admit frames of exponential functions in their L^2 spaces, establishing a generalized spectral property for these measures.

Contribution

It demonstrates that all locally and uniformly α-dimensional measures on α-quasi-regular sets have exponential frames, including the longstanding case of the middle-third Cantor measure.

Findings

Existence of exponential frames for the middle-third Cantor measure.

Generalization of spectral measures to fractal supports.

Answer to a long-standing open question in fractal harmonic analysis.

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 $L^2(\mu)$ 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 $L^2(\mu_C)$ (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 $L^2(\mu_C)$.