On Zeros of Certain Analytic Functions

Bolotnikov, Vladimir

doi:10.1007/s00020-010-1826-3

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The Blaschke Interpolation Problem and Rational γ-Inner Func1

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2017

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“…The following is a refinement of the Sarason Interpolation Theorem [40], in that we consider interpolation nodes both on the circle and in the open disc. The result is contained in [18,Theorem 2.5]. See also [20,Theorem 5.2] for a solution to the analogous interpolation problem for the upper half plane.…”

Section: The Blaschke Interpolation Problem and Rational γ-Inner Funcmentioning

confidence: 99%

Finite Blaschke products and the construction of rational Γ-inner functions

Agler

Lykova

Young

2017

Journal of Mathematical Analysis and Applications

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A Γ-inner function is a holomorphic map h from the unit disc D to Γ whose boundary values at almost all points of the unit circle T belong to the distinguished boundary bΓ of Γ. A rational Γ-inner function h induces a continuous map h| T from T to bΓ. The latter set is topologically a Möbius band and so has fundamental group Z. The degree of h is defined to be the topological degree of h| T . In a previous paper the authors showed that if h = (s, p) is a rational Γ-inner function of degree n then s 2 − 4p has exactly n zeros in the closed unit disc D − , counted with an appropriate notion of multiplicity. In this paper, with the aid of a solution of an interpolation problem for finite Blaschke products, we explicitly construct the rational Γ-inner functions of degree n with the n zeros of s 2 − 4p prescribed.

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Section: The Blaschke Interpolation Problem and Rational γ-Inner Funcmentioning

confidence: 99%