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DISCRETE APPROXIMATION OF NON-COMPACT OPERATORS DESCRIBING CONTINUUM-OF-ALLELES MODELS

Published online by Cambridge University Press:  01 July 2004

Oliver Redner
Oliver Redner
Affiliation:
Institut für Mathematik und Informatik, Universität Greifswald, Jahnstr. 15a, 17487 Greifswald, Germany ([email protected])
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Abstract

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We consider the eigenvalue equation for the largest eigenvalue of certain kinds of non-compact linear operators given as the sum of a multiplication and a kernel operator. It is shown that, under moderate conditions, such operators can be approximated arbitrarily well by operators of finite rank, which constitutes a discretization procedure. For this purpose, two standard methods of approximation theory, the Nyström and the Galerkin method, are generalized. The operators considered describe models for mutation and selection of an infinitely large population of individuals that are labelled by real numbers, commonly called continuum-of-alleles models.

AMS 2000 Mathematics subject classification: Primary 47A58; 45C05. Secondary 47B34

Keywords

continuum-of-alleles modelsmutation–selection modelsdiscretizationkernel operatorsmultiplication operatorsnon-compact operators
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 47 , Issue 2 , June 2004 , pp. 449 - 472
Copyright
Copyright © Edinburgh Mathematical Society 2004