Topics in Interpolation Theory of Rational Matrix-valued Functions

Topics in Interpolation Theory of Rational Matrix-valued Functions

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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 , q q Z/ are the given zeros with given multiplicates nl, q q n / and Wbq q 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.Suppose (E, G, g, f, s ) is a c. s. s. d. oven Q_ 60%. 3 om 2 stationa aamp; n x n math ix {unction W which Ap8 steguApasu on T. Then S^ aamp; a to $ 64. 23 the SyAdv 2A test 24 uation (2.15) sa#39; G a€“ gsa#39; = fB. Let us say that a collection of matrices {(C, A) , (a, b)anbsp;...


Title:Topics in Interpolation Theory of Rational Matrix-valued Functions
Author: I. Gohberg
Publisher:Birkhäuser - 2013-11-21
ISBN-13:

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