We know that $\{e^{2 \pi i n t}\}_{n \in \mathbb{Z}}$ is an orthonormal basis for $L^2[0, 1]$. Suppose I have a union of two disjoint intervals $I_1 \cup I_2$. Consider $L^2(I_1 \cup I_2)$, does this necessarily have an exponential orthonormal basis as above? Keep in mind, I do not mean we necessarily take the same set, but something of the form $\{e^{2 \pi i \alpha_n t}\}_{\alpha_n \in A}$. Can this be done? If not what are the known results, specifically for orthonormal basis?
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