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Orthogonal and similar polynomials: properties and applications


Applications of the theory of orthogonal polynomials are diffused among all types of problems of Pure and applied mathematics. These polynomials provide important tools for the solution of many problems and, for example, they have been contributing in studies related to differential equations, continued fractions, numerical stability, fast and super fast algorithms. Their applications varies from number theory to approximation theory, from combinatorics to group representations, from quantum mechanics to statistical physics and from system theory to signal processing (see for example the NATO Conference Proceedings "Nevai, P. (Ed.), Orthogonal Polynomials: theory and practice, Kluwer, Boston, 1990" for a good overview). An interesting application of a special class of orthogonal polynomials, known as the Jacobi polynomials, appears as a fundamental part of the proof of the Bieberbach Conjecture (of 1916) given by Louis de Branges in 1984. Our research group (GruPOsjrp), based in UNESP, Campus of São José do Rio Preto, has made many contributions to this theory and also to the theories of orthogonal and similar polynomials in relation to other inner products. The principal members of the group are Alagacone Sri Ranga, Cleonice Fátima Bracciali, Dimitar Kolev Dimitrov and Eliana Xavier Linhares de Andrade. Recent contributions of the group have covered studies of 1) orthogonal polynomials on real intervals, as defined above; 2) Szego polynomials, which are defined on the unit circle; 3) orthogonal Laurent polynomials, these are orthogonal functions which require measures with negative moments also; and 4) Sobolev orthogonal polynomials defined in terms of inner products involving derivatives. It is very gratifying to see that this large area of research, which attracted the curiosity of many famous mathematicians of the past, has turned out to be a very active area of interest. The interest of research in this area is so large, there is even a "SIAM activity group on Orthogonal Polynomials and Special Functions". The interest of GruPOsjrp is to have a significant participation in these research activities and by this, not only bringing benefits to the group through new publications, transmitting the acquired knowledge to our young students to form a new generation of good Brazilian Mathematicians. (AU)

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Scientific publications (6)
(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)
DIMITROV, DIMITAR K.; KOSTOV, VLADIMIR P.. Sharp Turan inequalities via very hyperbolic polynomials. Journal of Mathematical Analysis and Applications, v. 376, n. 2, p. 385-392, . (03/01874-2)
DIMITROV, DIMITAR K.; MARCELLAN, FRANCISCO; RAFAELI, FERNANDO R.. Monotonicity of zeros of Laguerre-Sobolev-type orthogonal polynomials. Journal of Mathematical Analysis and Applications, v. 368, n. 1, p. 80-89, . (03/01874-2)
DIMITROV, DIMITAR K.; MELLO, MIRELA V.; RAFAELI, FERNANDO R.. Monotonicity of zeros of Jacobi-Sobolev type orthogonal polynomials. APPLIED NUMERICAL MATHEMATICS, v. 60, n. 3, p. 263-276, . (03/01874-2)
DIMITROV, DIMITAR K.; KOSTOV, VLADIMIR P.. Distances between critical points and midpoints of zeros of hyperbolic polynomials. BULLETIN DES SCIENCES MATHEMATIQUES, v. 134, n. 2, p. 196-206, . (03/01874-2)
DIMITROV, DIMITAR K.; LUCAS, FABIO R.. HIGHER ORDER TURAN INEQUALITIES FOR THE RIEMANN xi-FUNCTION. Proceedings of the American Mathematical Society, v. 139, n. 3, p. 1013-1022, . (03/01874-2)
DIMITROV, DIMITAR K.; NIKOLOV, GENO P.. Sharp bounds for the extreme zeros of classical orthogonal polynomials. Journal of Approximation Theory, v. 162, n. 10, SI, p. 1793-1804, . (03/01874-2)

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