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A Lie-theoretic setting for the classical interpolation theories

A Lie-theoretic setting for the classical interpolation theories The three classical interpolation theories — Newton-Lagrange, Thiele and Pick-Nevanlinna — are developed within a common Lie-theoretic framework. They essentially involve a recursive process, each step geometrically providing an analytic map from a Riemann surface to a Grassmann manifold. The operation which passes from the (n−1)st to the nth involves the action of what the physicists call a group of gauge transformations. There is also a first-order difference operator which maps the set of solutions of the nth order interpolation to the (n−1)st: This difference operator is, in each case, covariant with respect to the action of the Lie groups involved. For Newton-Lagrange interpolation, this Lie group is the group of affine transformations of the complex plane; for Thiele interpolation the group SL(2, C) of projective transformations; and for Pick-Nevanlinna interpolation the subgroup SU(1, 1) of SL(2, C) which leaves invariant the disk in the complex plane. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

A Lie-theoretic setting for the classical interpolation theories

Hermann, Robert
Acta Applicandae Mathematicae , Volume 6 (3) – May 3, 2004
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References (25)

Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Computational Mathematics and Numerical Analysis; Applications of Mathematics; Partial Differential Equations; Probability Theory and Stochastic Processes; Calculus of Variations and Optimal Control; Optimization
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/BF00047160
Publisher site
See Article on Publisher Site

Abstract

The three classical interpolation theories — Newton-Lagrange, Thiele and Pick-Nevanlinna — are developed within a common Lie-theoretic framework. They essentially involve a recursive process, each step geometrically providing an analytic map from a Riemann surface to a Grassmann manifold. The operation which passes from the (n−1)st to the nth involves the action of what the physicists call a group of gauge transformations. There is also a first-order difference operator which maps the set of solutions of the nth order interpolation to the (n−1)st: This difference operator is, in each case, covariant with respect to the action of the Lie groups involved. For Newton-Lagrange interpolation, this Lie group is the group of affine transformations of the complex plane; for Thiele interpolation the group SL(2, C) of projective transformations; and for Pick-Nevanlinna interpolation the subgroup SU(1, 1) of SL(2, C) which leaves invariant the disk in the complex plane.

Journal

Acta Applicandae Mathematicae – Springer Journals

Published: May 3, 2004

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