The spectrality of symmetric additive measures

Abstract

where the component measure μ is the Lebesgue measure supported on [t,t+1] for t∈ℚ∖{-1 2} and δ 0 is the Dirac measure at 0. We prove that ρ is a spectral measure if and only if t∈1 2ℤ. In this case, L 2 (ρ) has a unique orthonormal basis of the form