We consider 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 ρ. 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$\{e^{2\pi i({\lambda }_1{x}_1+{\lambda }_2{x}_2)}:({\lambda }_1,{\lambda }_2)\in \mathrm{\Lambda }\},$ where$\mathrm{\Lambda }=\{(n/2,-n/2)|n\in \mathbb{Z}\}.$

我们考虑加性空间，它由两个单位长度的间隔或两个一般概率测度组成${\mathbb{[R}}^{\mathrm{1个}}$，位于 ${\mathbb{[R}}^{\mathrm{2个}}$，具有自然加法测度ρ。我们研究了指数框架，Riesz基和正交基的正态基之间的关系。${\mathrm{大号}}^{\mathrm{2个}}（\rho ）$及其组成空间。我们发现指数基础的存在在很大程度上取决于我们如何将度量定位于${\mathbb{[R}}^{\mathrm{1个}}$。我们证明非重叠加性空间具有Riesz基数，并且给出了重叠空间的必要条件。我们还表明，Lebesgue类型的一些重叠加性空间具有指数正交正态基，而另一些则没有。一个特殊的例子是在原点处的L形，它在形式的平移之前具有唯一的正交基础$\{{Ë}^{\mathrm{2个}\pi \mathrm{一世}（{\lambda }_{\mathrm{1个}}{X}_{\mathrm{1个}}+{\lambda }_{\mathrm{2个}}{X}_{\mathrm{2个}}）}：（{\lambda }_{\mathrm{1个}}，{\lambda }_{\mathrm{2个}}）\in \mathrm{\Lambda }\}，$ 在哪里$\mathrm{\Lambda }=\{（ñ/\mathrm{2个}，-ñ/\mathrm{2个}）|ñ\in \mathbb{ž}\}。$