  Intrinsic wavelet transforms and wavelet estimation methods are introduced
for curves in the non-Euclidean space of Hermitian positive definite matrices,
with in mind the application to Fourier spectral estimation of multivariate
stationary time series. The main focus is on intrinsic average-interpolation
wavelet transforms in the space of positive definite matrices equipped with an
affine-invariant Riemannian metric, and convergence rates of linear wavelet
thresholding are derived for intrinsically smooth curves of Hermitian positive
definite matrices. In the context of multivariate Fourier spectral estimation,
intrinsic wavelet thresholding is equivariant under a change of basis of the
time series, and nonlinear wavelet thresholding is able to capture localized
features in the spectral density matrix across frequency, always guaranteeing
positive definite estimates. The finite-sample performance of intrinsic wavelet
thresholding is assessed by means of simulated data and compared to several
benchmark estimators in the Riemannian manifold. Further illustrations are
provided by examining the multivariate spectra of trial-replicated brain signal
time series recorded during a learning experiment.
