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 R d and let μ and ν be two positive Borel measures on R d such that the convolution μ ∗ ν is a multiple of χ 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 R 1 and we show that it implies the classical Fuglede’s Conjecture on R 1 .