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On Quadratic and Bilinear Forms in Function Theory
Author(s) -
Fitzgerald Carl H.,
Horn Roger A.
Publication year - 1982
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-44.3.554
Subject(s) - mathematics , bilinear interpolation , analytic function , interpolation (computer graphics) , quadratic equation , generalization , bounded function , symmetric bilinear form , class (philosophy) , quadratic function , pure mathematics , characterization (materials science) , bilinear form , mathematical analysis , statistics , geometry , animation , materials science , computer graphics (images) , artificial intelligence , computer science , nanotechnology
We study questions concerning analyticity and analytic interpolation which can be formulated in terms of quadratic and bilinear inequalities. A theory of quadratic inequalities involving fixed number of terms is given, and sufficient conditions for analyticity are derived which lead to a generalization of a classical theorem of Pick and Nevanlinna on bounded analytic functions in the unit disc. Using elementary constuctive methods we present a theory of quadratic ineqalites involving arbitrarily many terms, and we use it to state sufficient conditions for the solutions of analytic interpolation problems in one and several dimensions. Finally we apply our major results to obtain a new characterization of the class of univalent bounded analytic functions in the unit disc. Our characterization leads to a new interpolation theorem for this class.