Orthogonal Bases of Exponential Functions for $$L^2(\mu )$$ on $$\mathbb {R}^d$$

A probability measure \(\mu \) on \({{\mathbb {R}}}^d\) with compact support is called a spectral measure if it possesses an exponential orthonormal basis for \(L^2(\mu )\). In this paper, we establish general criteria for determining whether a probability measure is spectral or not. As applications of these criteria, we provide a straightforward proof for the Lebesgue measure restricted to \([0, 1]^d\) or \([0, 1]\cup [a, a+1]\cup [b, b+1]\) to be a spectral measure. Furthermore, we investigate the spectrality of Cantor–Moran measure

$$\begin{aligned} \mu _{\{A_n, {{\mathcal {D}}}_n\}}= \delta _{A_1^{-1}{{\mathcal {D}}}_1}*\delta _{A_1^{-1}A_2^{-1}{{\mathcal {D}}}_2}*\delta _{A_1^{-1}A_2^{-1}A_3^{-1}{{\mathcal {D}}}_3}*\cdots \end{aligned}$$

generated by an admissible sequence \(\{(A_n,{{\mathcal {D}}}_n)\}_{n=1}^{\infty }\). It is noteworthy that our general criteria can be applied to establish numerous known and novel results.