  The subject of the present study is the Monte Carlo path-integral evaluation
of the moments of spectral functions. Such moments can be computed by formal
differentiation of certain estimating functionals that are
infinitely-differentiable against time whenever the potential function is
arbitrarily smooth. Here, I demonstrate that the numerical differentiation of
the estimating functionals can be more successfully implemented by means of
pseudospectral methods (e.g., exact differentiation of a Chebyshev polynomial
interpolant), which utilize information from the entire interval $(-\beta \hbar
/ 2, \beta \hbar/2)$. The algorithmic detail that leads to robust numerical
approximations is the fact that the path integral action and not the actual
estimating functional are interpolated. Although the resulting approximation to
the estimating functional is non-linear, the derivatives can be computed from
it in a fast and stable way by contour integration in the complex plane, with
the help of the Cauchy integral formula (e.g., by Lyness' method). An
interesting aspect of the present development is that Hamburger's conditions
for a finite sequence of numbers to be a moment sequence provide the necessary
and sufficient criteria for the computed data to be compatible with the
existence of an inversion algorithm. Finally, the issue of appearance of the
sign problem in the computation of moments, albeit in a milder form than for
other quantities, is addressed.
