| Abstract: |
| Given a finite Borel measure $\mu$ on $\Bbb{R}^d$, we give a characterization on the existence of a phase function $\varphi$ such that $L^2(\mu)$ admits an orthogonal basis/a frame/a Riesz basis of exponential type $E(\Lambda,\varphi)=\{e^{2\pi i \langle\lambda,\varphi(x)\rangle}:\lambda\in\Lambda\},$ according to the type of $\mu$: discrete, singularly and absolutely continuous. We show that if $\mu$ is an infinite discrete measure or a finite discrete measure with non-equal probability weights, then $L^2(\mu)$ can not admit any $E(\Lambda,\varphi)$ as an orthogonal basis. We also prove that $L^2(\mu)$ can admit some $E(\Lambda,\varphi)$ as an orthogonal basis if $\mu$ is one of the following four types
(i) a self-affine measure generated by an equal probability IFS with no-overlap condition;
(ii) the Lebesgue measure restricted on a bounded open set ;
(iii) a positive and finite absolutely continuous measure with respect to Lebesgue measure on $\Bbb{R}$;
(iv) a finite discrete measure with equal probability weights.
Particularly, for the product of two finite Borel measures $\mu\subseteq \Bbb{R}^m$ and $\nu\subseteq\Bbb{R}^n$, we study the relationship on the existence of orthogonal bases in the form of $E(\Lambda,\varphi)$ between in $L^2(\mu\times\nu)$ and in $L^2(\mu)$ and $L^2(\nu)$ according to whether $\Lambda$ or $\varphi$ has a product structure (i.e. $\Lambda=\Lambda_1\times\Lambda_2$ or $\varphi=\varphi_1\times\varphi_2$) or not. |
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