E-Thesis 373 views 186 downloads
Generalised Weyl algebras and Segal-Bargmann transform for the Meixner class of orthogonal polynomials / CHADAPHORN KODSUEB
Swansea University Author: CHADAPHORN KODSUEB
DOI (Published version): 10.23889/SUthesis.69061
Abstract
Meixner (1934) proved that there exist exactly five classes of orthogonal Sheffer sequences: Hermite polynomials that are orthogonal with respect to the Gaussian distribution, Char-lier polynomials orthogonal with respect to Poisson distribution, Laguerre polynomials orthogonal with respect to gamma...
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Swansea, Wales, UK
2024
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| Institution: | Swansea University |
| Degree level: | Doctoral |
| Degree name: | Ph.D |
| Supervisor: | Lytvynov, Eugene ; Finkelshtein, Dmitri |
| URI: | https://cronfa.swan.ac.uk/Record/cronfa69061 |
| Abstract: |
Meixner (1934) proved that there exist exactly five classes of orthogonal Sheffer sequences: Hermite polynomials that are orthogonal with respect to the Gaussian distribution, Char-lier polynomials orthogonal with respect to Poisson distribution, Laguerre polynomials orthogonal with respect to gamma distribution, Meixner polynomials of the first kind orthogonal with respect to negative binomial (Pascal) distribution, and Meixner polyno-mials of the second kind (or Meixner–Pollaczak polynomials) orthogonal with respect to Meixner distribution. Bargmann (1961) constructed a Hilbert space of entire functions on the complex plane, called nowadays Fock or Segal–Bargmann space. In this space, the cre-ation and annihilation operators act as multiplication by the variable and differentiation, respectively. These operators generate a Weyl algebra. The Segal–Bargmann transform provides a unitary isomorphism between the L2-space of the Gaussian distribution and the Fock space. This construction was later extended to the case of the Poisson distribution. The present dissertation deals with the latter three sets of orthogonal Sheffer sequences: Laguerre and Meixner of both the first and the second kind. We discuss generalised Weyl algebras that are naturally associated with these polynomials. By using a set of nonlinear coherent states, we construct a generalised Segal–Bargmann transform which is a unitary isomorphism between the L2-space of the orthogonality measure and a certain Fock space of entire functions on the complex plane. In a special case, such a Fock space was already studied by Alpay–Jørgensen–Seager–Volok (2013) and Alpay–Porat (2018). |
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| Item Description: |
ORCiD identifier: https://orcid.org/0000-0001-7937-6147 |
| Keywords: |
Segal-Bargmann transform, nonlinear coherent states, generalized Weyl algebra, gamma distribution, negative binomial distribution, Meixner distribution, Laguerre polynomials, Meixner polynomials |
| College: |
Faculty of Science and Engineering |
| Funders: |
Doctoral Training Program (DTP), UKRI (EPSRC - EP/T51798/1) |