  We discuss the choice of weight functions for the moments of the spectral
density for two-point correlators of hadronic currents over a finite energy
interval. Of phenomenological relevance is an analysis of the spectra of tau
lepton decay on the energy interval [0,M_tau] and the low-energy hadron
production in e+ e- annihilation. General arguments are given for the
calculability of such moments in perturbation theory using both a finite-order
analysis and infinite resummation on the contour. Nonperturbative contributions
emerging from the operator product expansion for two-point correlators are
discussed within explicit models for the physical spectra. The quantitative
analysis strongly disfavours weight functions that suppress the high-energy
contribution to theoretical moments. This is in agreement with expectations
from qualitative considerations in perturbative QCD with asymptotic freedom. We
discuss the implication of our results for the ultimate accuracy that can be
reached in tau decays and low-energy e+ e- annihilation into hadrons with
present experimental data.
