Topics in Interpolation Theory of Rational Matrix-Valued Functions

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 zjs 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.