Resumen de Mathematical framework for pseudo-spectra of linear stochastic difference equations

Andrés Bujosa Brun , Marcos Bujosa Brun, Antonio García Ferrer

• Although spectral analysis of stationary stochastic processes has solid mathematical foundations, this is not always so for the non-stationary case. Here, we establish a sound mathematical framework for the spectral analysis of non-stationary solutions of linear stochastic difference equations. To achieve it, the classical problem is embedded in a wider framework, the Rigged Hilbert space; the Fourier Transform is extended, and a new Extended Fourier Transform pair pseudocovariance function/pseudo-spectrum is defined. Our approach is an extension proper of the classical spectral analysis, where the Fourier Transform pair auto-covariance function/spectrum is a particular case, and consequently spectrum and pseudo-spectrum are identical when the first one is defined.