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

Gohberg, I.

ISBN 10: 3034854714 / ISBN 13: 9783034854719

Editore: Birkhäuser, 2014

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Dati bibliografici

• Titolo: Topics in Interpolation Theory of Rational Matrix-valued Functions (Operator Theory: Advances and Applications)
• Editore: Birkhäuser
• Data di pubblicazione: 2014
• Lingua: Inglese
• ISBN 10: 3034854714
• ISBN 13: 9783034854719
• Rilegatura: Brossura
• Condizione: New

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Riassunto

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.