Interpolation by holomorphic maps from the disc to the tetrablock

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:498

Article

Alshammari, Hadi O.¹  Lykova, Zinaida A.² 

[1] Jouf Univ, King Khaled Rd, Skaka, Saudi Arabia

[2] Newcastle Univ, Sch Math Stat & Phys, Newcastle Upon Tyne NE1 7RU, Tyne & Wear, England

关键词: Blaschke product;    Tetrablock;    Inner functions;    Interpolation;    Pick matrix;    Distinguished boundary;   

DOI  :  10.1016/j.jmaa.2021.124951

来源: Elsevier

【 摘 要 】

The tetrablock is the set epsilon = {x is an element of C-3 : 1- x(1)z -x(2)w + x(3)zw not equal 0 whenever vertical bar z vertical bar <= 1, vertical bar w vertical bar <= 1}. The closure of epsilon is denoted by (epsilon) over bar. A tetra-inner function is an analytic map x from the unit disc D to (epsilon) over bar such that, for almost all points lambda of the unit circle T, lim(r up arrow 1) x(r lambda) exists and lies in b (epsilon) over bar, where b (epsilon) over bar denotes the distinguished boundary of (epsilon) over bar. There is a natural notion of degree of a rational tetra-inner function x; it is simply the topological degree of the continuous map x vertical bar T from T to b (epsilon) over bar. In this paper we give a prescription for the construction of a general rational tetra-inner function of degree n. The prescription exploits a known construction of the finite Blaschke products of given degree which satisfy some interpolation conditions with the aid of a Pick matrix formed from the interpolation data. It is known that if x = (x(1), x(2), x(3)) is a rational tetra-inner function of degree n, then x(1) x(2) - x(3) either is identically 0 or has precisely n zeros in the closed unit disc (D) over bar, counted with multiplicity. It turns out that a natural choice of data for the construction of a rational tetra-inner function x = (x(1), x(2), x(3)) consists of the points in (D) over bar for which x(1) x(2) - x(3) = 0 and the values of x at these points. (C) 2021 The Author(s). Published by Elsevier Inc.

【 授权许可】

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