Topics in Interpolation Theory of Rational Matrix-valued Functions (Operator Theory: Advances and Applications .33) （Softcover reprint of the original 1st ed. 1988. 2014. ix, 247 S. IX, 2）

Gohberg, I.

SPRINGER, BASEL; BIRKHÄUSER BASEL; BIRKHÄUSER（2014発売）

SPRINGER, BASEL; BIRKHÄUSER BASEL; BIRKHÄUSER（2014発売）

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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 , " " 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.