  In this paper, we study the problem of adaptive estimation of the spectral
density of a stationary Gaussian process. For this purpose, we consider a
wavelet-based method which combines the ideas of wavelet approximation and
estimation by information projection in order to warrants that the solution is
a nonnegative function. The spectral density of the process is estimated by
projecting the wavelet thresholding expansion of the periodogram onto a family
of exponential functions. This ensures that the spectral density estimator is a
strictly positive function. Then, by Bochner's theorem, the corresponding
estimator of the covariance function is semidefinite positive. The theoretical
behavior of the estimator is established in terms of rate of convergence of the
Kullback-Leibler discrepancy over Besov classes. We also show the excellent
practical performance of the estimator in some numerical experiments.
