Generalized moments of spectral functions from short-time correlation functions

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