Algebraic approach in pseudo-spectra estimation

We prove that $m$ principal singular vectors of a matrix $X_{d}$ constructed on the basis of a time series, contained periodical deterministic components with additive white noise, have equal pseudo-spectra and their pseudo-spectral structure is identical to that of the time series. The structures of pseudo-spectra of the rest singular vectors differ from the structures of pseudo-spectra of the principal vectors and the time series. It is shown that the time series allow one to increase the resolving capacity and to improve the statistical stability of spectral estimation.