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Let $Q$ be a fundamental domain of some full-rank lattice in ${\Bbb R}^d$ andlet $\mu$ and $\nu$ be two positive Borel measures on ${\Bbb R}^d$ such thatthe convolution $\mu\ast\nu$ is a multiple of $\chi_Q$. We consider the problemas to whether or not both measures must be spectral (i.e. each of theirrespective 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 alarge class of examples of spectral measures which are either absolutelycontinuous, singularly continuous or purely discrete spectral measures. Inaddition, we propose a generalized Fuglede's conjecture for spectral measureson ${\Bbb R}^1$ and we show that it implies the classical Fuglede's conjectureon ${\Bbb R}^1$.

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