Wronskians of Fourier and Laplace transforms

Abstract

Associated with a given suitable function, or a measure, on R \mathbb {R} , we introduce a correlation function so that the Wronskian of the Fourier transform of the function is the Fourier transform of the corresponding correlation function, and the same holds for the Laplace transform. We obtain two types of results. First, we show that Wronskians of the Fourier transform of a nonnegative function on R \mathbb {R} are positive definite functions and that the Wronskians of the Laplace transform of a nonnegative function on R + \mathbb {R}_+ are completely monotone functions. Then we establish necessary and sufficient conditions in order that a real entire function, defined as a Fourier transform of a positive kernel K K , belongs to the Laguerre–Pólya class, which answers an old question of Pólya. The characterization is given in terms of a density property of the correlation kernel related to K K , via classical results of Laguerre and Jensen and employing Wiener’s L 1 L^1 Tauberian theorem. As a consequence, we provide a necessary and sufficient condition for the Riemann hypothesis in terms of a density of the translations of the correlation function related to the Riemann ξ \xi -function.