Ortogonal polynomials on the real line and on the unit circle

• Grant number: 12/11228-0
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: September 01, 2012
• End date: August 31, 2014
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Alagacone Sri Ranga
• Grantee: Mauro Matushima
• 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

It is very gratifying to see that this area of research which has kindled the interest of many famous mathematicians of the past, has once again turned out to be an extremely active area of research. The interest of the research group in UNESP, São José do Rio Preto continues to have an active part in this new surge of research activities, not only benefitting the group with new publications but also passing the acquired knowledge to new participants of the group, thus helping in the formation of a new generation of good young mathematics. In the coming years, however, the study of orthogonal polynomials on the unit circle will have priority in the research activities of the member Alagacone Sri Ranga of the group. These polynomials are also commonly known as Szegö polynomials in honor of Gábor Szegö who introduced them in the second half of the 20th century. Because of their applications in quadrature rules, signal processing, operator and spectral theory, and many other topics, these polynomials have received a lot of attention in recent years. Very recently, we have observed that any sequence of orthogonal polynomials on the unit circle can also be characterized in terms of a pair of real sequences, where one of the sequences is also a positive chain sequence. This observation opens up a new window to look into the fascinating world of these polynomials. Thus, the main objective of this project is to study the properties of orthogonal polynomials on the unit circle starting from these sequences. The objective of the Iniciação Científica work involved in this project is to systematically provide the student with the knowledge that is necessary for him to be able carryout research in this área.(AU)