Generalisane kvadraturne formule Gauss-ovog tipa

Abstract

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.