Given a domain Ω⊂Rd\documentclass[12pt]{minimal}
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\begin{document}$$\varOmega \subset {\mathbb {R}}^d$$\end{document} with positive and finite Lebesgue measure and a discrete set Λ⊂Rd\documentclass[12pt]{minimal}
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\begin{document}$$\varLambda \subset {\mathbb {R}}^d$$\end{document}, we say that (Ω,Λ)\documentclass[12pt]{minimal}
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\begin{document}$$(\varOmega , \varLambda )$$\end{document} is a frame spectral pair if the set of exponential functions E(Λ):={e2πiλ·x:λ∈Λ}\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {E}}(\varLambda ):=\{e^{2\pi i \lambda \cdot x}: \lambda \in \varLambda \}$$\end{document} is a frame for L2(Ω)\documentclass[12pt]{minimal}
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\begin{document}$$L^2(\varOmega )$$\end{document}. Special cases of frames include Riesz bases and orthogonal bases. In the finite setting ZNd\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_N^d$$\end{document}, d,N≥1\documentclass[12pt]{minimal}
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\begin{document}$$d, N\ge 1$$\end{document}, a frame spectral pair can be similarly defined. In this paper we show how to construct and obtain new classes of frame spectral pairs in Rd\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^d$$\end{document} by “adding” a frame spectral pair in Rd\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^{d}$$\end{document} to a frame spectral pair in ZNd\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {Z}}_N^d$$\end{document}. Our construction unifies the well-known examples of exponential frames for the union of cubes with equal volumes. We also remark on the link between the spectral property of a domain and sampling theory.
