Finite eigenfuction approximations for continuous spectrum operators
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...
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| Format: | Article |
| Language: | English |
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Wiley
1993-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171293000018 |
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| _version_ | 1832564237401063424 |
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| author | Robert M. Kauffman |
| author_facet | Robert M. Kauffman |
| author_sort | Robert M. Kauffman |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-0392ea7397094d0fb5ad67ee92f268af |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1993-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-0392ea7397094d0fb5ad67ee92f268af2025-02-03T01:11:29ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251993-01-0116112210.1155/S0161171293000018Finite eigenfuction approximations for continuous spectrum operatorsRobert M. Kauffman0Department of Mathematics, University of Alabama at Birmingham, Birmingham, Alabama 35294, USAIn 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.http://dx.doi.org/10.1155/S0161171293000018continuous spectrum eigenfunction expansionself-adjoint operatorordinary differential operatorpartial differential operatorspectral theorem. |
| spellingShingle | Robert M. Kauffman Finite eigenfuction approximations for continuous spectrum operators International Journal of Mathematics and Mathematical Sciences continuous spectrum eigenfunction expansion self-adjoint operator ordinary differential operator partial differential operator spectral theorem. |
| title | Finite eigenfuction approximations for continuous spectrum operators |
| title_full | Finite eigenfuction approximations for continuous spectrum operators |
| title_fullStr | Finite eigenfuction approximations for continuous spectrum operators |
| title_full_unstemmed | Finite eigenfuction approximations for continuous spectrum operators |
| title_short | Finite eigenfuction approximations for continuous spectrum operators |
| title_sort | finite eigenfuction approximations for continuous spectrum operators |
| topic | continuous spectrum eigenfunction expansion self-adjoint operator ordinary differential operator partial differential operator spectral theorem. |
| url | http://dx.doi.org/10.1155/S0161171293000018 |
| work_keys_str_mv | AT robertmkauffman finiteeigenfuctionapproximationsforcontinuousspectrumoperators |