Minimal symmetric Darlington synthesis

We consider the symmetric Darlington synthesis of a p x p rational symmetric Schur function S with the constraint that the extension is of size 2p x 2p. Under the assumption that S is strictly contractive in at least one point of the imaginary axis, we determine the minimal McMillan degree of the extension. In particular, we show that it is generically given by the number of zeros of odd multiplicity of I-SS*. A constructive characterization of all such extensions is provided in terms of a symmetric realization of S and of the outer spectral factor of I-SS*. The authors's motivation for the problem stems from Surface Acoustic Wave filters where physical constraints on the electro-acoustic scattering matrix naturally raise this mathematical issue

On the theory of matrix-valued functions belonging to the Smirnov class

A theory of matrix-valued functions from the matricial Smirnov class N + n (D) is systematically developed. In particular, the maximum principle of V.I.Smirnov, inner-outer factorization, the Smirnov-Beurling characterization of outer functions and an analogue of Frostman’s theorem are presented for matrix-valued functions from the Smirnov class N + n (D). We also consider a family Fλ = F − λI of functions belonging to the matricial Smirnov class which is indexed by a complex parameter λ. We show that with the exception of a ”very small ” set of such λ the corresponding inner factor in the inner-outer factorization of the function Fλ is a Blaschke-Potapov product. The main goal of this paper is to provide users of analytic matrix-function theory with a standard source for references related to the matricial Smirnov class. NOTATIONS: C- the complex plane. T: = {t ∈ C: |t | = 1}- the unit circle. D: = {z ∈ C: |z | < 1}- the unit disc. BT- the σ- algebra of Borel subsets of T. m- normalized Lebesgue measure on the measurable space (T, BT). Cn- the n-dimensional complex space equipped with the usual Euclidean norm, i.e., for x = (ξ1,...,ξn) ⊤ we define ‖x‖Cn: