No Cover Image

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...

Full description

Published: Swansea, Wales, UK 2024
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
Supervisor: Lytvynov, Eugene ; Finkelshtein, Dmitri
URI: https://cronfa.swan.ac.uk/Record/cronfa69061
first_indexed 2025-03-07T11:14:08Z
last_indexed 2025-03-08T05:55:20Z
id cronfa69061
recordtype RisThesis
fullrecord <?xml version="1.0"?><rfc1807><datestamp>2025-03-07T11:21:16.6480728</datestamp><bib-version>v2</bib-version><id>69061</id><entry>2025-03-07</entry><title>Generalised Weyl algebras and Segal-Bargmann transform for the Meixner class of orthogonal polynomials</title><swanseaauthors><author><sid>4f3034e19f0877ddf0b7ddf83e6cc7e8</sid><firstname>CHADAPHORN</firstname><surname>KODSUEB</surname><name>CHADAPHORN KODSUEB</name><active>true</active><ethesisStudent>false</ethesisStudent></author></swanseaauthors><date>2025-03-07</date><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&#x2013;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&#x2013;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&#x2013;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&#x2013;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&#x2013;J&#xF8;rgensen&#x2013;Seager&#x2013;Volok (2013) and Alpay&#x2013;Porat (2018).</abstract><type>E-Thesis</type><journal/><volume/><journalNumber/><paginationStart/><paginationEnd/><publisher/><placeOfPublication>Swansea, Wales, UK</placeOfPublication><isbnPrint/><isbnElectronic/><issnPrint/><issnElectronic/><keywords>Segal-Bargmann transform, nonlinear coherent states, generalized Weyl algebra, gamma distribution, negative binomial distribution, Meixner distribution, Laguerre polynomials, Meixner polynomials</keywords><publishedDay>12</publishedDay><publishedMonth>11</publishedMonth><publishedYear>2024</publishedYear><publishedDate>2024-11-12</publishedDate><doi>10.23889/SUthesis.69061</doi><url/><notes>ORCiD identifier: https://orcid.org/0000-0001-7937-6147</notes><college>COLLEGE NANME</college><CollegeCode>COLLEGE CODE</CollegeCode><institution>Swansea University</institution><supervisor>Lytvynov, Eugene ; Finkelshtein, Dmitri</supervisor><degreelevel>Doctoral</degreelevel><degreename>Ph.D</degreename><degreesponsorsfunders>Doctoral Training Program (DTP), UKRI (EPSRC - EP/T51798/1)</degreesponsorsfunders><apcterm/><funders>Doctoral Training Program (DTP), UKRI (EPSRC - EP/T51798/1)</funders><projectreference/><lastEdited>2025-03-07T11:21:16.6480728</lastEdited><Created>2025-03-07T11:08:03.6895242</Created><path><level id="1">Faculty of Science and Engineering</level><level id="2">School of Mathematics and Computer Science - Mathematics</level></path><authors><author><firstname>CHADAPHORN</firstname><surname>KODSUEB</surname><order>1</order></author></authors><documents><document><filename>69061__33759__876d2a7f9755491092f915a192a69b8f.pdf</filename><originalFilename>Kodsueb_Chadaphorn_PhD_Thesis_Final_Cronfa.pdf</originalFilename><uploaded>2025-03-07T11:18:09.7791624</uploaded><type>Output</type><contentLength>970022</contentLength><contentType>application/pdf</contentType><version>E-Thesis &#x2013; open access</version><cronfaStatus>true</cronfaStatus><documentNotes>Copyright: The Author, Chadaphorn Kodsueb, 2024.</documentNotes><copyrightCorrect>true</copyrightCorrect><language>eng</language></document></documents><OutputDurs/></rfc1807>
spelling 2025-03-07T11:21:16.6480728 v2 69061 2025-03-07 Generalised Weyl algebras and Segal-Bargmann transform for the Meixner class of orthogonal polynomials 4f3034e19f0877ddf0b7ddf83e6cc7e8 CHADAPHORN KODSUEB CHADAPHORN KODSUEB true false 2025-03-07 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). E-Thesis Swansea, Wales, UK Segal-Bargmann transform, nonlinear coherent states, generalized Weyl algebra, gamma distribution, negative binomial distribution, Meixner distribution, Laguerre polynomials, Meixner polynomials 12 11 2024 2024-11-12 10.23889/SUthesis.69061 ORCiD identifier: https://orcid.org/0000-0001-7937-6147 COLLEGE NANME COLLEGE CODE Swansea University Lytvynov, Eugene ; Finkelshtein, Dmitri Doctoral Ph.D Doctoral Training Program (DTP), UKRI (EPSRC - EP/T51798/1) Doctoral Training Program (DTP), UKRI (EPSRC - EP/T51798/1) 2025-03-07T11:21:16.6480728 2025-03-07T11:08:03.6895242 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics CHADAPHORN KODSUEB 1 69061__33759__876d2a7f9755491092f915a192a69b8f.pdf Kodsueb_Chadaphorn_PhD_Thesis_Final_Cronfa.pdf 2025-03-07T11:18:09.7791624 Output 970022 application/pdf E-Thesis – open access true Copyright: The Author, Chadaphorn Kodsueb, 2024. true eng
title Generalised Weyl algebras and Segal-Bargmann transform for the Meixner class of orthogonal polynomials
spellingShingle Generalised Weyl algebras and Segal-Bargmann transform for the Meixner class of orthogonal polynomials
CHADAPHORN KODSUEB
title_short Generalised Weyl algebras and Segal-Bargmann transform for the Meixner class of orthogonal polynomials
title_full Generalised Weyl algebras and Segal-Bargmann transform for the Meixner class of orthogonal polynomials
title_fullStr Generalised Weyl algebras and Segal-Bargmann transform for the Meixner class of orthogonal polynomials
title_full_unstemmed Generalised Weyl algebras and Segal-Bargmann transform for the Meixner class of orthogonal polynomials
title_sort Generalised Weyl algebras and Segal-Bargmann transform for the Meixner class of orthogonal polynomials
author_id_str_mv 4f3034e19f0877ddf0b7ddf83e6cc7e8
author_id_fullname_str_mv 4f3034e19f0877ddf0b7ddf83e6cc7e8_***_CHADAPHORN KODSUEB
author CHADAPHORN KODSUEB
author2 CHADAPHORN KODSUEB
format E-Thesis
publishDate 2024
institution Swansea University
doi_str_mv 10.23889/SUthesis.69061
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description 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).
published_date 2024-11-12T05:25:58Z
_version_ 1851369517016416256
score 11.089572

Similar Items