Abstract
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{e2πi(λ1x1+λ2x2):(λ1,λ2)∈Λ}, whereΛ={(n/2,−n/2)|n∈Z}.
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