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Let \(Q\) be a fundamental domain of some full-rank lattice in \({\mathbb {R}}^d\) and let \(\mu \) and \(\nu \) be two positive Borel measures on \({\mathbb {R}}^d\) such that the convolution \(\mu *\nu \) 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 \({\mathbb {R}}^1\) and we show that it implies the classical Fuglede’s Conjecture on \({\mathbb {R}}^1\).

Spectral Measures Associated with the Factorization of the Lebesgue Measure on a Set via Convolution