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arXiv · 1812.05708

A Fourier Frame for the Middle-Third Cantor Measure

Carlos Cabrelli·Ursula Molter
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Abstract

In this paper we show that if $μ$ is any locally and uniformly $α$-dimensional measure supported on a $α$-quasi-regular set $E$, then $L^2(μ)$ admits a frame of exponentials. In particular, for the uniform middle third Cantor measure, $μ_C,$ our result shows that there exists a countable set $Λ$ such that $\{e^{2πi t λ}\}_{λ\in Λ}$ is a frame for $L^2(μ_C)$ (i.e. the measure $μ_C$ admits a generalized spectrum), answering an old outstanding question about the existence of a frame of exponentials for the space $L^2(μ_C)$.

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Carlos Cabrelli, Ursula Molter. 2018-12-18. A Fourier Frame for the Middle-Third Cantor Measure. https://arxiv.org/abs/1812.05708

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