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Spectral Theory and Special Functions Contents. 1. Introduction 2. the Spectral Theorem 2.1. Hilbert Spaces and Bounded Operators 2.2. the Spectral Theorem for Bounded Self-adjoint Operators 2.3. Unbounded Self-adjoint Operators 2.4. the Spectral Theorem for Unbounded Self-adjoint Operators 3. Ortho
| Content Provider | Semantic Scholar |
|---|---|
| Author | Koelink, Erik |
| Copyright Year | 2008 |
| Abstract | 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(Z≥0), 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(Z). We discuss the theorem of Masson and Repka linking the deficiency indices of a Jacobi operator on l(Z) to those of two Jacobi operators on l(Z≥0). For two examples of Jacobi operators on l(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. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://arxiv.org/pdf/math/0107036v1.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
