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VOL. 44 | 2010 Algebraic Operators, Divided Differences, Functional Calculus, Hermite Interpolation and Spline Distributions
Sergey Ajiev

Editor(s) Andrew Hassell, Alan McIntosh, Robert Taggart


This article combines three components corresponding to real analysis, algebra and operator theory. A simple and universal way of finding a Hermite interpolation (quasi)polynomial and Gel'fond's formula covering both real and complex cases and based on the study of the sufficient conditions for the continuity of divided differences with respect to two types of the simultaneous convergence of nodes is presented. Relying on it, we establish both algebraic and topological Jordan decomposition for algebraic operators and algebraic and topological properties of their calculus, such as a constructive algebraic characterization of the solvability and an explicit estimate of the related a priori constant and a new characterization of bounded algebraic operators in terms of orbits. Further applications include explicit relations between the barycentric or uniform distributions on convex polyhedra and, correspondingly, $B$-splines or Steklov splines and short analytic proofs of some classical results from the polynomial arithmetics, wavelet theory and discrete Fourier transform. We also provide correct deffinitions, typical properties and representations for the holomorphic calculus of the closed operators with mixed spectra including both (double)sectorial and bounded components.


Published: 1 January 2010
First available in Project Euclid: 18 November 2014

zbMATH: 1236.47013
MathSciNet: MR2655390

Rights: Copyright © 2010, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.


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