A spectral characterization of the behavior of discrete time AR-representations over a finite time interval

Abstract

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Construction of Algebraic and Difference Equations with a Prescribed Solution Space

“…(Antoniou et al, 1998;Gohberg et al, 2009) An eigenpair of the dual matrixÃ(σ) corresponding to the eigenvalueλ = 0 is called an infinite eigenpair of A(σ) (or an infinite Jordan pair). Taking an eigenpair for each finite zeroλ = 0 ofÃ(σ), we construct the infinite spectral pair of A(σ),…”

“…(Antoniou et al, 1998;Gohberg et al, 2009) Let A(σ) be as in (3). Let also n and μ be the sums of the degrees of the finite and infinite elementary divisors of A(σ), respectively, as defined in (7) and (14).…”

“…(Antoniou et al, 1998;Gohberg et al, 2009) Let (C i ∈ C r×ni , J i ∈ C ni×ni ) be a matrix pair, where J i is in Jordan form, corresponding to the zero of λ i ∈ C of A(σ) with multiplicity n i . That is, J i consists of Jordan blocks with sizes equal to the partial multiplicities of λ i .…”

“…(Antoniou et al, 1998) The backward solution space B B of (2) is given by the column span of the matrix…”

“…The number q is often called the lag of studied by various authors (Antoniou et al, 1998;Antsaklis and Michel, 2006;Gohberg et al, 2009;Karampetakis, 2004). A method for constructing the forward (resp.…”