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We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special R II recurrence relation. We also look into some methods for generating the nodes (which lie on the real line) and the positive weights of these quadrature rules. With a simple transformation these quadrature rules on the real line also lead to certain positive quadrature rules of highest algebraic degree of precision on the unit circle. This way, we also introduce new approaches to evaluate the nodes and weights of these specific quadrature rules on the unit circle.
We consider the theoretical and numerical aspects of the quadrature rules associated with a sequence of polynomials generated by a special R II recurrence relation. We also look into some methods for generating the nodes (which lie on the real line) and the positive weights of these quadrature rules. With a simple transformation these quadrature rules on the real line also lead to certain positive quadrature rules of highest algebraic degree of precision on the unit circle. This way, we also introduce new approaches to evaluate the nodes and weights of these specific quadrature rules on the unit circle.
OrthoQuad 2014 was an international symposium on orthogonality, quadrature, and related topics, held on January 20-24, 2014 in Puerto de la Cruz, Tenerife, Spain. It was held in memory of Professor Pablo González-Vera (1955-2012. A short CV of Pablo González-VeraPablo González-Vera was born in Vallehermoso (La Gomera, Canary Islands) on January 25, 1955. He studied Mathematics at University of La Laguna (1974-79), and began teaching at this university in 1980 and became Cátedratico (Full Professor) de Matemática Aplicada in 1992.He started his research in rational interpolation and Padé approximation, in collaboration with Professor Luis Casasús (now in Universidad Politécnica de Madrid) [1,3]. In particular Padé-type and two-point Padé approximation got his specific interest [4-7, 9-14, 20].This resulted in a Doctoral Dissertation on Two-point Padé Approximation [8] that he defended in 1985 at the University of La Laguna. His supervisor was Professor Nácere E. Hayek Calil.But soon orthogonality of the denominator polynomials of the approximants and Laurent polynomials got his interest and these showed up in another application, which became his most favored topic: numerical quadrature [2,[15][16][17][18][19][20].That was the germ of a research group on Approximation Theory in University of La Laguna. From 1991 on, eight doctoral thesis on Orthogonality and/or Quadrature were supervised by González-Vera, with the most recent being defended shortly after his passing away.Since 1989 a collaboration of the "gang of four" started: Adhemar Bultheel (KU Leuven, Belgium), Erik Hendriksen (University of Amsterdam, The Netherlands) and Olav Njåstad (University of Trondheim, Norway). In 1990 a local report of 84 pages [21] was published that formed the embryo of what became ten years later their book Orthogonal Rational Functions (Cambridge University Press, 1999) [110].During these ten years they collaborated intensively on the properties of these rational functions that generalized the orthogonal polynomials and Laurent polynomials. The support of the orthogonality measure could be the unit circle or on the real line (the whole real half line, the half line or a finite interval). They appeared in multipoint versions of Padé approximation, in generalizations of classical moment problems, and in quadrature formulas on the real line or the unit circle to generalize Gauss and Szegő numerical integration methods. [37, 38, 46-49, 61, 62, 66-73, 81-83, 87, 88, 96-99, 102-105, 108, 109]. Their investigations were deepened further in the subsequent period [117-119, 126-128, 132, 138-140, 142, 145, 154, 164, 173-175, 179, 183].With his PhD students he started several parallel research topics like Laurent Padé and Padé-type approximation (M. Camacho, PhD 1991) [51,52,74,100,101], multipoint Padé-type approximation and quadrature (M. Jiménez Paíz, PhD 1991) [15, 27, 30, 32, 41, 64, 80, 90, 112, 115], multivariate Padé approximation (R. Orive, PhD 1991), [42,59,60] composite and alternative numerical quadrature formulas (J...
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