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

A probability measure \(\mu \) $\mu$ on \({{\mathbb {R}}}^d\) ${\mathbb{R}}^d$ with compact support is called a spectral measure if it possesses an exponential orthonormal basis for \(L^2(\mu )\) $L^2(\mu )$ . 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 \([0, 1]^d\) ${[0,1]}^d$ or \([0, 1]\cup [a, a+1]\cup [b, b+1]\) $[0,1]\cup [a,a+1]\cup [b,b+1]$ to be a spectral measure. Furthermore, we investigate the spectrality of Cantor–Moran measure \(\begin{aligned} \mu _{\{A_n, {{\mathcal {D}}}_n\}}= \delta _{A_1^{-1}{{\mathcal {D}}}_1}*\delta _{A_1^{-1}A_2^{-1}{{\mathcal {D}}}_2}*\delta _{A_1^{-1}A_2^{-1}A_3^{-1}{{\mathcal {D}}}_3}*\cdots \end{aligned}\) $$\begin{array}{c}\hfill {\mu }_{\{A_n,{\mathcal{D}}_n\}}={\delta }_{A_1^{-1}{\mathcal{D}}_1}\ast {\delta }_{A_1^{-1}{A}_2^{-1}{\mathcal{D}}_2}\ast {\delta }_{A_1^{-1}{A}_2^{-1}{A}_3^{-1}{\mathcal{D}}_3}\ast \cdots \end{array}$$ generated by an admissible sequence \(\{(A_n,{{\mathcal {D}}}_n)\}_{n=1}^{\infty }\) ${\{(A_n,{\mathcal{D}}_n)\}}_{n=1}^{\infty }$ . It is noteworthy that our general criteria can be applied to establish numerous known and novel results.