Spectral properties of some unions of linear spaces

We consider additive spaces, consisting of two intervals of unit length or two general probability measures on R, positioned on the axes in R, with a natural additive measure ρ. We study the relationship between the exponential frames, Riesz bases, and orthonormal bases of L(ρ) and those of its component spaces. We find that the existence of exponential bases depends strongly on how we position our measures on R. We show that non-overlapping additive spaces possess Riesz bases, and we give a necessary condition for overlapping spaces. We also show that some overlapping additive spaces of Lebesgue type have exponential orthonormal bases, while some do not. A particular example is the L shape at the origin, which has a unique orthonormal basis up to translations of the form {e:(λ,λ)∈Λ}, where Λ={(n/2,−n/2)|n∈Z}.