Quadrature rules for periodic integrands

In this paper, the algebraic construction of quadrature formulas for weigh- ted periodic integrals is revised. For this purpose, two classical papers ([10] and [14]) in the literature are revisited and certain relations and connections are brought to light. In this respect, the concepts of “bi-orthogonality” and “para-orthogonality” are shown to play a fundamental role. Key Words: Trigonometric polynomials, Szegő polynomials, quadratures, bi-orthogonality, para-orthogonality. AMS Classification Number: 41A55, 33C4

Bi-ortogonalidad y fórmulas de cuadratura positivas para integrandos periódicos.

El objetivo principal de esta Memoria es la construcci´on y caracterizaci´on de f´ormulas de cuadratura positivas para integrandos peri´odicos con respecto a una funci´on peso. Para ello, tomaremos como referencia base el art´ıculo [2], que fue inspirado a su vez por el c´elebre art´ıculo [6] escrito por G. Szeg˝o en 1963, y en donde los sistemas bi-ortogonales de polinomios trigonom´etricos son introducidos por primera vez en la literatura. Este concepto ser´a esencial para el c´alculo aproximado de integrales pesadas con integrandos peri´odicos.The main purpose of this Memory is the construction and characterization of positive quadrature formulas for periodic integrands with respect to a weight function. For this aim, we will take as basic reference the paper [2], which was actually inspired by the famous paper [6] written by G. Szeg˝o in 1963, and where the bi-orthogonal systems of trigonometric polynomials were introduced for the first time in the literature. This concept will be essential for the approximate calculation of weighted integrals with periodic integrands

Studies in numerical quadrature

Various types of quadrature formulae for oscillatory integrals are studied with a view to improving the accuracy of existing techniques. Concentration is directed towards the production of practical algorithms which facilitate the efficient evaluation of integrals of this type arising in applications. [Continues.

Generalisane kvadraturne formule Gauss-ovog tipa

Ova doktorska disertacija je zapravo deo rezultata, objedinjenih u celinu, do- bijenih tokom višegodišnjeg rada pod mentorstvom profesora Gradimira V. Milo- vanovića. Oblast istraživanja u okviru ove disertacije je razmatranje nekih nes- tandardnih tipova ortogonalnosti i njihova primena na konstrukciju kvadraturnih formula maksimalnog stepena tačnosti. S jedne strane radi se o istraživanjima povezanim sa Teorijom ortogonalnih sistema, što po prirodi pripada Teoriji apro- ksimacija, a sa druge strane konstrukciji kvadraturnih formula za numeričku in- tegraciju funkcija, kao važnom delu Numeričke analize. Moja istraživanja u ovoj oblasti započeta su izradom magistarske teze "Nes- tandardne ortogonalnosti i odgovaraju¶ce kvadrature Gauss-ovog tipa" ([86]), a u okviru projekta "Primenjeni ortogonalni sistemi, konstruktivne aproksimacije i numerički metodi" (Finansiranog od strane Ministarstva nauke i zaštite životne sredine Republike Srbije u periodu 2002{2005). U tom periodu publikovano je nekoliko radova ([65]{[67], [87]), u kojima su objedinjena tri različita pravca istraživanja: ortogonalnost na polukrugu u kompleksnoj ravni u odnosu na ne- hermitski skalarni proizvod, koncept s-ortogonalnosti, kao i koncept višestruke ortogonalnosti.The theory and applications of integration is one of the most important and central themes of mathematics. According to this fact, the subject Numerical Integration is one of the basic in numerical analysis. The problem of numerical integration is open-ended, no finite collection of techniques is likely to cover all possibilities that arise and to which an extra bit of special knowledge may be of great assistance. The field of research in this dissertation is consideration of some nonstandard types of orthogonality and their applications to constructions of quadrature rules with maximal degree of exactness, i.e., quadrature rules of Gaussian type. The research in this dissertation is connected with the following subjects: Theory of Orthogonality, Numerical Integration and Approximation Theory. We have tried to produce a balanced work between theoretical results and numerical algorithms. Gauss's famous method of approximate integration from 1814 can be extended in the several ways. In this dissertation, two ways of possible generalizations are considered. The first, a natural way, is an extension to non-polynomial functions. The second way is a generalization to quadrature rules with multiple nodes. These two generalizations are connected with some systems of trigonometric functions. Also, Gaussian type quadratures for some systems of fast oscillatory functions are considered