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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Complex interpolation of normed and quasinormed spaces in several dimensions. II. Properties of harmonic interpolation


Author: Zbigniew Slodkowski
Journal: Trans. Amer. Math. Soc. 317 (1990), 255-285
MSC: Primary 46M35; Secondary 32A30, 32F05, 46A99, 46B70
MathSciNet review: 949900
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Abstract: This paper is a continuation of the study of harmonic interpolation families of normed or quasinormed spaces parametrized by points of a domain in $ {{\mathbf{C}}^k}$. It is shown, among other things, that each of the following properties holds for all the intermediate quasinormed spaces, if it holds for all given boundary spaces: (1) being a normed space; (2) being a Hilbert space; (3) satisfying the triangle inequality by the $ r$th power of the quasinorm; (4) being uniformly convex; and (5) being uniformly smooth. As a principal tool, the notion of a harmonic set valued function (a generalization of analytic multifunction) is introduced and studied.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0949900-2
PII: S 0002-9947(1990)0949900-2
Article copyright: © Copyright 1990 American Mathematical Society