Topics in Interpolation Theory of Rational Matrix-valued Functions (Operator Theory: Advances and Applications, 33)

One of the basic interpolation problems from our point of view is the problem of building a scalar rational function if its poles and zeros with their multiplicities are given. If one assurnes that the function does not have a pole or a zero at infinity, the formula which solves this problem is (1) where Zl , " " Z/ are the given zeros with given multiplicates nl, " " n / and Wb" " W are the given p poles with given multiplicities ml, . . . ,m , and a is an arbitrary nonzero number. p An obvious necessary and sufficient condition for solvability of this simplest Interpolation pr- lern is that Zj :f: wk(1~ j ~ 1, 1~ k~ p) and nl +. . . +n/ = ml +. . . +m ' p The second problem of interpolation in which we are interested is to build a rational matrix function via its zeros which on the imaginary line has modulus 1. In the case the function is scalar, the formula which solves this problem is a Blaschke product, namely z z. )mi n u(z) = all = l~ (2) J ( Z+ Zj where [o] = 1, and the zj's are the given zeros with given multiplicities mj. Here the necessary and sufficient condition for existence of such u(z) is that zp :f: - Zq for 1~ ]1, q~ n. Front Matter....Pages I-IX Realization and Interpolation of Rational Matrix Functions....Pages 1-72 Interpolation Problems for Rational Matrix Functions with Incomplete Data and Wiener-Hopf Factorization....Pages 73-108 Regular Rational Matrix Functions with Prescribed Pole and Zero Structure....Pages 109-122 Inverse Spectral Problems for Regular Improper Rational Matrix Functions....Pages 123-173 Unitary Rational Matrix Functions....Pages 175-222 Proper Contractions and their Unitary Minimal Completions....Pages 223-247 Back Matter....Pages N1-N1