We consider additive spaces, consisting of two intervals of unit length or two general probability measures on R-1, positioned on the axes in R-2, with a natural additive measure rho. We study the relationship between the exponential frames, Riesz bases, and orthonormal bases of L-2(rho) and those of its component spaces. We find that the existence of exponential bases depends strongly on how we position our measures on R-1. 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(2 pi i(lambda 1x lambda + lambda 2x2) :) (lambda(1), x(2)) : (lambda 1, lambda 2) is an element of Lambda}, where Lambda = {( n/2,- n/2) n is an element of Z}. (C) 2021 Elsevier Inc. All rights reserved.
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