Numerical integration is the study of how the numerical value of an integral
can be found. Also called quadrature, which refers to finding a square whose area
is the same as the area under a curve, it is one of the classical topics of numerical
analysis. Of the central interest is the process of approximating a definite integral
from values of the integrand when exact mathematical integration is not available.
The principal topic of this doctoral dissertation is Gaussian quadrature rule
with maximal trigonometric degree of exactness, as a generalization of the classical
Gaussian quadrature rule for algebraic polynomials. The research in this
dissertation is connected with the following subjects: Theory of Orthogonality,
Numerical Integration and Approximation Theory.
This dissertation, beside Preface and References with 75 items, consists of
five chapters: Introduction; Orthogonal systems of trigonometric polynomials;
Quadrature rules of Gaussian type for trigonometric poly...nomials; Error estimates
for quadrature rules of Gaussian type for trigonometric polynomials; Multiple orthogonal
trigonometric polynomials of semi–integer degree and the corresponding
quadrature rules.
The first chapter of the thesis contains a short history of Gaussian quadrature
rules for algebraic polynomials, including some error estimates of such quadrature
rules and their generalizations on non–polynomial functions. Generalization on
quadrature rules for trigonometric polynomials is a motivation for the research
that is presented within this dissertation.
Definitions and features of trigonometric polynomials of semi–integer and integer
degree are presented in the second chapter.
The third chapter is devoted to quadrature rules with maximal trigonometric
degree of exactness, i.e., quadrature rules of Gaussian type. The quadrature
rules with an even and an odd number of nodes are observed particulary. Also,
in the both cases (even and odd numbers of nodes) quadrature rules with even weight functions are considered and the connection with corresponding Gaussian
quadrature rules for algebraic polynomials is given.
The fourth chapter presents the latest results about error estimates for Gaussian
quadrature rule for trigonometric polynomials. At first, error estimates in the
case of quadrature rules with an odd numbers of nodes for 2π–periodic functions,
analytic in circular domain, and with respect to the weight functions w(x) = 1,
x ∈ [0, 2π), w(x) = 1 + cos x and w(x) = 1 − cos x, x ∈ [−π, π) are given. Also,
for some even weight functions the error bounds of Gaussian quadrature rules
for trigonometric polynomials in the both cases (even and odd number of nodes)
and for 2π–periodic integrand in certain domain of complex plane (circular and
elliptic) are given. Several numerical examples are also included.
In the fifth chapter multiple orthogonal trigonometric polynomials of semi–
integer degree and the corresponding optimal quadrature formulae for trigonometric
polynomials are introduced. Also, some numerical examples are included.
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.
