Spectral measures associated with the factorization of the Lebesgue measure on a set via convolution

Let $Q$ be a fundamental domain of some full-rank lattice in ${\Bbb R}^d$ and let $\mu$ and $\nu$ be two positive Borel measures on ${\Bbb R}^d$ such that the convolution $\mu\ast\nu$ is a multiple of $\chi_Q$. We consider the problem as to whether or not both measures must be spectral (i.e. each of their respective associated $L^2$ space admits an orthogonal basis of exponentials) and we show that this is the case when $Q = [0,1]^d$. This theorem yields a large class of examples of spectral measures which are either absolutely continuous, singularly continuous or purely discrete spectral measures. In addition, we propose a generalized Fuglede's conjecture for spectral measures on ${\Bbb R}^1$ and we show that it implies the classical Fuglede's conjecture on ${\Bbb R}^1$.