On spectral factorization in multidimensions (Q2702497)

The author constructs an algorithm for obtaining a solution to the scalar version (\(m=1\)) of the following factorization problem, the matricial version of which (\(m\geq 1\)) was considered first in [\textit{A. Kummert}, Multidimensional Syst. Signal Process. 1, No. 3, 327-339 (1991; Zbl 0725.93020)] where the existence of a solution has been proved. Let \(\varphi\) be an (\(m\times m\)) polynomial matrix in two variables \(p=(p_1,p_2)\) with normal rank \(r\) such that (i) \(\varphi\) is para-Hermitian (i.e., \(\varphi_*=\varphi\), where \(\varphi_*(p_1,p_2):=\varphi(-\overline{p}_1,-\overline{p}_2)^*\)), (ii) \(\varphi\) is nonnegative definite for all \(p=j\omega\), where \(\omega =(\omega_1,\omega_2)\in {\mathbb R} ^2\). Find a (\(2r\times m\)) rational matrix \(H\) which is holomorphic in Re \(p>0\) and is a spectral factor of \(\varphi\) in the sense \(\varphi =H_*H\). Moreover, find such an \(H\) which is polynomial in one of the variables, say \(p_2\), and is rational in the other variable \(p_1\). The result can be reformulated as a representation of a positive polynomial in two variables as a sum of squares of rational functions. In the author's construction of such a representation, the Pfister forms from abstract algebra are used.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00062].