Neke modifikacije klasičnih mera, odgovarajući ortogonalni polinomi i kvadrature Gausovog tipa

Abstract

Sažetak: Predmet ove disertacije jesu polinomi realne promenljive ortogonalni u odnosu na polinomske modifikacije Čebiševljevih mera prve i druge vrste, njihova simboliqka izraqunavanja i diferencne osobine. Pored ovoga, predmet disertacije jeste i efikasan numerički algoritam za konstrukciju parametara generalizovanih Gaus-Risovih kvadraturnih formula i numerička konstrukcija odgovarajuih klasa ortogonalnih polinoma. Za polinome ortogonalne u odnosu na modifikovane Čebiševljeve mere prve i druge vrste određeni su koeficijenti tročlane rekurentne relacije u simboličkom obliku i ispitivane su njihove diferencno-diferencijabilne osobine. Pored ovoga, pretpostavljajući logaritamski potencijal data je elektrostatiqka interpretacija nula polinoma ortogonalnih u odnosu na modifikovanu Čebiševljevu meru prve vrste.

Abstract: In this thesis we consider the polynomials of real variable orthogonal with respect to the polynomial modifications of the Chebyshev measures of the first and second kind, its symbolic calculations and difference properties. In addition, the subject of this dissertation is an efficient numerical algorithm for the construction of parameters of the generalized Gauss-Rys quadrature formulas and numerical construction of the corresponding orthogonal polynomials. For polynomials orthogonal with respect to the modified Chebyshev measures of the first and second kind, the coefficients of the three-term recurrence relation are obtained in their symbolic form and also the difference properties are examined. Furthermore, assuming a logarithmic potential, we gave an electrostatic interpretation of the zeros of the polynomials orthogonal with respect to the modified Chebyshev measure of the first kind. In a recent paper by Milovanovi´c [48], an efficient numerical algorithm for the construction of Guass-Rys quadratures was given. In this study, the so-called generalized Gauss-Rys quadrature formulas with respect to the product of the exponential function exp (−xt2 ) and the ultraspherical Gegenbauer weight function are studied, with a special attention to the important Chebyshev cases. All obtained results have been checked in the software Mathematica with intensive use of the package OrthogonalPolynomials.