Integration and Laplace Transformation of Orthogonal Series

Abstract

<p>In the first part of this thesis the basic properties of classical orthogonal polynomials are derived from a unified theory. Then sufficient conditions for term-by-term integration and Laplace transformation of Fourier expansions in terms of an orthogonal system with respect to a weight function on a bounded or unbounded interval are given, and successively applied to Fourier expansions in terms of the trigonometric system, the classical orthogonal polynomials, the Haar system and eigenfunction expansions for ordinary linear differential equations with boundary conditions on both ends of a compact interval as well as in limit point and limit circle cases for an infinite interval. Finally a necessary and sufficient condition for representation of a complex function as a Laplace interval of f ε L(0,2π) with period 2π is proved.</p>