Spectral theory and special functions
| dc.creator | Koelink, Erik | |
| dc.date | 2001-07-05 | |
| dc.date.accessioned | 2026-07-07T04:42:29Z | |
| dc.date.available | 2026-07-07T04:42:29Z | |
| dc.description | A short introduction to the use of the spectral theorem for self-adjoint operators in the theory of special functions is given. As the first example, the spectral theorem is applied to Jacobi operators, i.e. tridiagonal operators, on l^2(N), leading to a proof of Favard's theorem stating that polynomials satisfying a three-term recurrence relation are orthogonal polynomials. We discuss the link to the moment problem. In the second example, the spectral theorem is applied to Jacobi operators on l^2(Z). We discuss the theorem of Masson and Repka linking the deficiency indices of a Jacobi operator on l^2(Z) to those of two Jacobi operators on l^2(N). For two examples of Jacobi operators on l^2(Z), namely for the Meixner, respectively Meixner-Pollaczek, functions, related to the associated Meixner, respectively Meixner-Pollaczek, polynomials, and for the second order hypergeometric q-difference operator, we calculate the spectral measure explicitly. This gives explicit (generalised) orthogonality relations for hypergeometric and basic hypergeometric series. | |
| dc.description | Lecture notes for the SIAM Activity Group OP-SF summer school 2000, Laredo, Spain. 40 page, latex | |
| dc.identifier | https://arxiv.org/abs/math/0107036 | |
| dc.identifier | http://arxiv.org/abs/math/0107036 | |
| dc.identifier | 45-84 in `Laredo Lectures on Orthogonal Polynomials and Special Functions' (eds. R. Álvarez-Nodarse, F. Marcellán, W. Van Assche), Nova Science Publishers, 2004. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/61802 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.title | Spectral theory and special functions | |
| dc.type | text |