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 R¹, positioned on the axes in R², with a natural additive measure ρ. We study the relationship between the exponential frames, Riesz bases, and orthonormal bases of L²(ρ) and those of its component spaces. We find that the existence of exponential bases depends strongly on how we position our measures on R¹. 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^{2πi(λ₁x₁+λ₂x₂)}:(λ₁,λ₂)∈Λ}, where Λ={(n/2,−n/2)|n∈Z}.