Polynomials obtained from a recurrence relation which is a modification of the rec...
Orthogonal and similar polynomials: properties and applications
| Grant number: | 14/08328-8 |
| Support Opportunities: | Research Grants - Visiting Researcher Grant - International |
| Start date: | August 14, 2014 |
| End date: | September 13, 2014 |
| Field of knowledge: | Physical Sciences and Mathematics - Mathematics - Applied Mathematics |
| Principal Investigator: | Dimitar Kolev Dimitrov |
| Grantee: | Dimitar Kolev Dimitrov |
| Visiting researcher: | Yuan Xu |
| Visiting researcher institution: | University of Oregon (UO), United States |
| Host Institution: | Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil |
| Associated research grant: | 09/13832-9 - Orthogonal polynomials, special functions and applications, AP.TEM |
Abstract
A recent paper of the proposer reveals another tight relation between Statistical Mechanics and Orthogonal Polynomials. The partition function in Statistical Mechanics can be viewed as a bilateral Laplace transform of a Borel measure. Therefore the properties of all significant models can be successfully studied via properties of the Laplace and Fourier transform. An interesting characterization of the so-called Lee-Yang measure, was given in terms of the Wronskian of the polynomials that are orthogonal with respect to the measure. Then the Wronskian may be symetrized to obtain multivariate polynomials that are symmetric with respect to its variables and posses some orthogonal properties. The new polynomials of many variables turn out to be wave function of a multi-particle quantum system. While this surprising interplay between fundamental notions in Statistical and Quantum mechanics, the partition function and the wave functions, is still to understood and interpreted properly from Physics point of view, the study of the multivariate symmetric polynomials, represented in a determinant form, composed by orthogonal polynomials, is of interest from mathematical point of view. Another interesting question which arises and will be discussed is the possibility of performing spectral analysis via the Fourier transform. The classical approach is via the Stieltjes transform of the corresponding Borel measure. The second topic, which will be discussed with members of the Group on Functional Analysis of the Department of Mathematics, at campus in São Carlos of The University of São Paulo, concerns positive definite functions. The São Carlos group is interested in extending some classical result of Pólya which hold for spheres of small dimension to any $n$-dimensional sphere. (AU)
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