  The nonparametric volatility estimation problem of a scalar diffusion process
observed at equidistant time points is addressed. Using the spectral
representation of the volatility in terms of the invariant density and an
eigenpair of the infinitesimal generator the first known estimator that attains
the minimax optimal convergence rates for both high and low-frequency
observations is constructed. The proofs are based on a posteriori error bounds
for generalized eigenvalue problems as well as the path properties of scalar
diffusions and stochastic analysis. The finite sample performance is
illustrated by a numerical example.
