  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 ${\Bbb R}^1$.
