A Paley-Wiener Type Theorem for Singular Measures on $\mathbb{T}$

Eric S. Weber

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arXiv · 1709.07522

A Paley-Wiener Type Theorem for Singular Measures on $\mathbb{T}$

Eric S. Weber

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Abstract

For a fixed singular Borel probability measure $\mu$ on $\mathbb{T}$, we give several characterizations of when an entire function is the Fourier transform of some $f \in L^2(\mu)$. The first characterization is given in terms of criteria for sampling functions of the form $\hat{f}$ when $f \in L^2(\mu)$. The second characterization is given in terms of criteria for interpolation of bounded sequences on $\mathbb{N}_{0}$ by $\hat{f}$. Both characterizations use the construction of Fourier series for $f \in L^2(\mu)$ demonstrated in Herr and Weber via the Kaczmarz algorithm and classical results concerning the Cauchy transform of $\mu$.

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BibTeX RIS

Eric S. Weber. 2017-09-21. A Paley-Wiener Type Theorem for Singular Measures on $\mathbb{T}$. https://arxiv.org/abs/1709.07522

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