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arXiv · 2510.23754

Some Plancherel identities for unbounded subsets of $\mathbb R$ in duality

Piyali Chakraborty·Dorin Ervin Dutkay
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Abstract

In relation to Fuglede's conjecture, we establish several Plancherel-type identities and demonstrate the surjectivity of the Fourier transform between certain unbounded tiling sets of $\mathbb{R}$ that are in duality. In the terminology commonly used in the context of Fuglede's conjecture, our result states that an open set tiles $\mathbb{R}$ by the finite set $\{0,1,\dots,p-1\}$ if and only if it admits a spectrum (or, equivalently, a dual pair measure) given by the Lebesgue measure on $\left[-\tfrac{1}{2p}, \tfrac{1}{2p}\right] + \mathbb{Z}$.

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Piyali Chakraborty, Dorin Ervin Dutkay. 2025-10-27. Some Plancherel identities for unbounded subsets of $\mathbb R$ in duality. https://arxiv.org/abs/2510.23754

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