On Fourier frame of absolutely continuous measures

Let mu be a compactly supported absolutely continuous probability measure on R-n, we show that L-2(K, d mu) admits a Fourier frame if and only if its Radon-Nikodym derivative is bounded above and below almost everywhere on the support K. As a consequence, we prove that if mu is an equal weight absolutely continuous self-similar measure on R-1 and L-2(K, d mu) admits a Fourier frame, then the density of mu must be a characteristic function of self-similar tile. In particular, this shows for almost everywhere 1/2 < lambda < 1, the L-2 space of the lambda-Bernoulli convolutions cannot admit a Fourier frame. (C) 2011 Elsevier Inc. All rights reserved.