International Journal of Mathematics and Mathematical Sciences
Volume 16 (1993), Issue 1, Pages 1-22
Finite eigenfuction approximations for continuous spectrum operators
Department of Mathematics, University of Alabama at Birmingham, Birmingham 35294, Alabama, USA
Received 23 January 1992
Copyright © 1993 Robert M. Kauffman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In this paper, we introduce a new formulation of the theory of continuous spectrum
eigenfunction expansions for self-adjoint operators and analyze the question of when operators
may be approximated in an operator norm by finite sums of multiples of eigenprojections of
multiplicity one. The theory is designed for application to ordinary and partial differential
equations; relationships between the abstract theory and differential equations are worked out in
the paper. One motivation for the study is the question of whether these expansions are
susceptible to computation on a computer, as is known to be the case for many examples in the
discrete spectrum case. The point of the paper is that continuous and discrete spectrum
eigenfunction expansions are treated by the same formalism; both are limits in an operator norm
of finite sums.