From the introduction: “Let \(\mu\) be a compactly supported Borel probability measure on \(\mathbb{R}^d\). We say that \(\mu\) is a frame spectral measure if there exists a collection of exponential functions \(\{e^{2\pi i\langle \lambda ,x\rangle}\}_{\lambda\in\Lambda}\) such that there exist \(0<A\leq B<\infty\) with \[ A\parallel f\parallel_2^2\leq\sum_{\lambda\in\Lambda}|\int f(x) e^{-2\pi i\langle \lambda ,x\rangle} d\mu(x)|^2 \leq B\parallel f\parallel_2^2 \;\;\;\;\text{for all} \;\;\;\; f\in L^2(\mu) . \] Whenever such a \(\Lambda\) exists, \(\{e^{2\pi i\langle \lambda ,x\rangle}\}_{\lambda\in\Lambda}\) is called a Fourier frame for \(L^2(\mu)\) and \(\Lambda\) is a frame spectrum for \(\mu\). When \(\mu\) admits an exponential orthonormal basis, we say that \(\mu\) is a spectral measure and the corresponding frequency set \(\Lambda \) is called a spectrum for \(\mu\).”
“It is natural to ask the following question:
(Q). Can a non-spectral fractal measure still admit some Fourier frame?
This question was possibly first proposed by R. S. Strichartz [J. Anal. Math. 81, 209–238 (2000; Zbl. 0976.42020), p. 212]. In particular, there have been discussions asking whether specifically the one-third Cantor measure can be frame spectral. Although we are unable to settle the case of the one-third Cantor measure, the main purpose of this paper is to answer (Q) positively with explicit examples.”