Generalized moments of spectral functions from short-time correlation functions

TL;DR

This paper introduces an integral transformation that extracts spectral moments from short-time correlation data, relying on analytic continuation, and provides conditions for its convergence and efficiency in spectral analysis.

Contribution

It presents a novel integral transformation method for obtaining spectral moments from short-time correlation functions using analytic continuation.

Findings

Converges for correlation functions with analytic continuation to the entire complex plane.

Valid for Paley-Wiener functions obtained by Fourier-Laplace transform of compactly supported distributions.

Allows exponential speed evaluation of moments when the distribution support is within a specific interval.

Abstract

We present an integral transformation capable of extracting moments of arbitrary Paley-Wiener entire functions against a given spectral distribution based solely on short-time values of the correlation function in a small open disk about the origin. The integral is proven to converge absolutely to the expected result for those correlation functions that can be extended analytically to the entire complex plane, with the possible exception of two branch cuts on the imaginary axis. It is only the existence of an analytic continuation that is required and not the actual values away from the small disk about the origin. If the analytic continuation exists only for a strip |Im(z)| < \tau_0, then the integral transformation remains valid for all Paley-Wiener functions obtained by Fourier-Laplace transforming a compactly supported distribution, with the support included in the interval (-2\tau_0, 2\tau_0). Finally, if the support of the distribution is contained in the interval $(-\tau_0, \tau_0)$, then the generalized moment can be evaluated from the short-time values of the correlation function exponentially fast