Adaptive covariance estimation of locally stationary processes

It is shown that the covariance operator of a locally stationary process ha s approximate eigenvectors that are local cosine functions. We model locall y stationary processes with pseudo-differential operators that are time-var ying convolutions. An adaptive covariance estimation is calculated by searc hing first for a "best" local cosine basis which approximates the covarianc e by a band or a diagonal matrix. The estimation is obtained from regulariz ed versions of the diagonal coefficients in the best basis.