Razvoj racionalnih algoritama za konstrukciju ortogonalnih polinoma jedne promenljive

Abstract

Orthogonal polynomials, their construction, analysis and application, have an important role in the Applied mathematics. Nevertheless, the investigations in the Constructive theory of Orthogonal polynomials are still in an early development phase. The central concept is a three-term recurrence relation. Knowing its coefficients is of outmost importance. Once these are known, all other computational aspects regarding Orthogonal polynomials are easily available. Obtaining these coefficients is a fairly difficult task, thus, keeping up-to-date libraries on new developments is of high importance. The new era mighty computer packages are being built containing the readily-available results and also the tools for calculating the yet unknown parameters. Orthogonal polynomials, as all other special functions, owe their origin and development to practice in our physical and the scientific world. As Mathematics develops and awareness in its application broadens, the new horizons open in different scientific fields. Such is the case with the Constructive theory of Orthogonal polynomials. The number of its applications is augmenting on daily bases as a result of new developments in the topic. Among applications with the far most importance and longest history is the one in Numerical integration. The underlying concept is orthogonal decomposition providing the least error with a prescribed computational effort. Thus the search for the most appropriate dense subspace of approximation. This dissertation is a result of the research I was involved in under the supervision and guidance of academician Gradimir Milovanovi´c. The research addressed the two key issues, mentioned earlier, related to the theory of Orthogonal polynomials on real line. The summary of the known and original results is given in a systematic overview through four chapters. The Bibliography contains 71 titles of scientific papers and books. The first chapter is introductory. Contains basic definitions and properties of the theory of Orthogonal polynomials. In particular, a separate Section is devoted to Chebyshev polynomials of the first and second kind. In the final section two algorithms for the construction of the three-term recurrence coefficients are enclosed. Modification algorithms are treated in the second chapter. Weight functions of the linear functional are transformed and the influence on the recurrence coefficients is observed. The original algorithm for the quadratic Christoffel modification is presented along with its application. Modification of Chebyshev’s measures of the first and second kind is the topic of the third chapter. Research of these non-classic weights finalized in the established recurrence coefficients for an entirely new class of orthogonal polynomials. The forth chapter is devoted to numerical integration. Necessary and sufficient conditions for the positive definiteness of certain class of linear functionals is presented. A standard L2-approximation is also treated. As an example, the modification algorithm from the second chapter is applied to a polynomial L2-approximation.