A Paley–Wiener Type Theorem for Singular Measures on \((-1/2, 1/2)\)

For a fixed singular Borel probability measure \(\mu \) on \((-1/2, 1/2)\), 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 by Herr and Weber via the Kaczmarz algorithm and classical results concerning the Cauchy transform of \(\mu \).