Discrete Approximation of Non-Compact Operators Describing Continuum-of-Alleles Models

Details Discrete Approximation of Non-Compact Operators Describing Continuum-of-Alleles Models Discrete Approximation of Non-Compact Operators Describing Continuum-of-Alleles Models

2003-01-03

arxiv.org/abs/math/0301024v2

Proc. Edinb. Math. Soc. (2004) 47, 449--472

Mathematics

Spectral Theory

22 pages; small changes

Scientific paper

10.1017/S0013091503000476

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 Nystr\"om and the Galerkin method, are generalized. The operators considered describe models for mutation and selection of an infinitely large population of individuals that are labeled by real numbers, commonly called continuum-of-alleles (COA) models.

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