On Fourier frame of absolutely continuous measures

Abstract

Let $\mu$ be a compactly supported absolutely continuous probability measure on ${\Bbb R}^n$, we show that $\mu$ admits Fourier frames if and only if its Radon-Nikodym derivative is upper and lower bounded almost everywhere on its support. As a consequence, we prove that if an equal weight absolutely continuous self-similar measure on ${\Bbb R}^1$ admits Fourier frame, then the measure must be a characteristic function of self-similar tile. In particular, this shows for almost everywhere $1/2<\lambda<1$, the $\lambda$-Bernoulli convolutions cannot admit Fourier frames.