Orthogonal polynomials on the unit circle and related studies

• Grant number: 17/12324-6
• Support Opportunities: Regular Research Grants
• Start date: September 01, 2017
• End date: August 31, 2018
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Alagacone Sri Ranga
• Grantee: Alagacone Sri Ranga
• 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 researchers: Cleonice Fátima Bracciali

Abstract

In the coming years, studies of orthogonal polynomials on the unit circle will have priority in the research activities of the member Alagacone Sri Ranga of gruPOSjrp of IBILCE/UNESP. These polynomials are also commonly known as Szegö polynomials in honor of Gábor Szeg\H{o} 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 $\{c_n\}_{n \geq 1}$ and $\{d_{n}\}_{n \geq 1}$, where $\{d_{n}\}_{n \geq 1}$ is also a positive chain sequence. This observation opens up a new window to look into the fascinating world of these polynomials. Thus, one of the main objectives of this project is to study the properties of orthogonal polynomials on the unit circle in terms of the sequences $\{c_n\}_{n \geq 1}$ and $\{d_{n}\}_{n \geq 1}$. New classes of orthogonal polynomials (also orthogonal polynomials of type Sobolev) on the unit circle that we have encountered recently have created new challenges to be confronted. Finally, as the most recent ``work front'' to study orthogonal polynomials on the unit circle, we present a generalized eigenvalue problem which should lead to many research papers in the coming future. (AU)

Scientific publications

(5)

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

ISMAIL, M. E. H.; SRI RANGA, A.. R-II type recurrence, generalized eigenvalue problem and orthogonal polynomials on the unit circle. Linear Algebra and its Applications, v. 562, p. 63-90, FEB 1 2018. (17/12324-6, 16/09906-0)

BRACCIALI, CLEONICE F.; SILVA, JAIRO S.; RANGA, A. SRI. Extended Relativistic Toda Lattice, L-Orthogonal Polynomials and Associated Lax Pair. ACTA APPLICANDAE MATHEMATICAE, v. 164, n. 1, p. 137-154, DEC 2019. (17/12324-6, 16/09906-0)

BRACCIALI, CLEONICE F.; PEREIRA, JUNIOR A.; RANGA, A. SRI. Quadrature rules from a R-II type recurrence relation and associated quadrature rules on the unit circle. NUMERICAL ALGORITHMS, v. 83, n. 3, p. 1029-1061, MAR 2020. (17/12324-6, 16/09906-0)

BRACCIALI, C. F.; MARTINEZ-FINKELSHTEIN, A.; SRI RANGA, A.; VERONESE, D. O.. Christoffel formula for kernel polynomials on the unit circle. Journal of Approximation Theory, v. 235, p. 46-73, NOV 2018. (17/12324-6, 16/09906-0)

MARTINEZ-FINKELSHTEIN, A.; SILVA RIBEIRO, L. L.; SRI RANGA, A.; TYAGLOV, M.. COMPLEMENTARY ROMANOVSKI-ROUTH POLYNOMIALS: FROM ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE TO COULOMB WAVE FUNCTIONS. Proceedings of the American Mathematical Society, v. 147, n. 6, p. 2625-2640, JUN 2019. (17/12324-6, 16/09906-0, 17/04358-8)