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Continuous analogs of Schur extension problems and bitangential generalizations
Author(s) -
Arov Damir Z.,
Dym Harry
Publication year - 2012
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201100262
Subject(s) - mathematics , holomorphic function , toeplitz matrix , extension (predicate logic) , unit disk , complex plane , schur's theorem , pure mathematics , combinatorics , matrix (chemical analysis) , triangular matrix , mathematical analysis , classical orthogonal polynomials , gegenbauer polynomials , materials science , invertible matrix , computer science , composite material , orthogonal polynomials , programming language
The classical Schur extension problem is to describe the set of contractive holomorphic functions s (ζ) in the open unit disk of the form\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty} $$ s(\zeta )=a_0+a_1\zeta +\cdots +a_n\zeta ^n+\cdots $$ \end{document} It is well‐known that this set is nonempty if and only if the ( n + 1) × ( n + 1) lower triangular Toeplitz matrix A with entries a ij = a i − j is contractive, i.e., if and only if I n + 1 − AA * ≥ 0; see e.g., p. 79 of 5. In this paper the analogue of this problem for mvf's (matrix valued functions) that are holomorphic and contractive in the open upper half plane and bitangential generalizations thereof are studied.