Constructing trends of time series segments

The sub-band analysis enabling one to construct a sequence whose Fourier transform is the best approximation of a segment of Fourier transform of the original series within a given frequency interval was shown to be an efficient tool to specify the trends of segments of the nonstationary time series. Relations were established defining the matrix operator to sort out such components. A procedure for adaptive construction of the operators for trend extraction was proposed, and conditions were determined under which a wide class of sequence segments are their eigenfunctions (fixed points) corresponding to the unit eigenvalues.