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
ope...n 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.
Faculty:
Универзитет у Крагујевцу, Природно-математички факултет
