Spectral properties of some unions of linear spaces

We consider \textit{additive spaces}, consisting of two intervals of unit length or two general probability measures on ${\mathbb R}^1$, positioned on the axes in ${\mathbb R}^2$, 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 ${\mathbb R}^1$. 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 \[ \left\{e^{2 \pi i (\lambda_1 x_1 + \lambda_2 x_2)} : (\lambda_1, \lambda_2) \in \Lambda \right\}, \] where \[ \Lambda = \{ (n/2, -n/2) \mid n \in {\mathbb Z} \}. \]