2024-04-12T18:11:41Zhttps://docta.ucm.es/rest/oai/requestoai:docta.ucm.es:20.500.14352/414682023-11-14T08:22:20Zcom_20.500.14352_14col_20.500.14352_17
Bujosa Brun, Marcos
Bujosa Brun, Andrés
García Ferrer, Antonio
2023-06-19T23:52:50Z
2023-06-19T23:52:50Z
2013
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https://hdl.handle.net/20.500.14352/41468
https://www.ucm.es/icae
Although spectral analysis of stationary stochastic processes has solid mathematical foundations, this is not always so for some non-stationary cases. Here, we establish a rigorous mathematical extension of the classic Fourier spectrum to the case in which there are AR roots in the unit circle, ie, the transfer function of the linear time-invariant filter has poles on the unit circle. To achieve it we: embed the classical problem in a wider framework, the Rigged Hilbert space, extend the Discrete Time Fourier Transform and defined a new Extended Fourier Transform pair pseudo-covariance function/pseudo-spectrum. Our approach is a proper extension of the classical spectral analysis, within which the Fourier Transform pair auto-covariance function/spectrum is a particular case. Consequently spectrum and pseudo-spectrum coincide when the first one is defined.
eng
open access
Mathematical framework for pseudo-spectra of linear stochastic difference equations
technical report