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Abstract
The different classes of classical orthogonal polynomials are nowadays part of the essential mathematical machinery of numerous spectral algorithms. While the essential properties and the fundamental approximation results related to the infinite orthogonal polynomials is long well know, these objects have not been derived for the finite classes of orthogonal polynomials and integrated into the spectral methods.The aim of this thesis is to present the essential properties and the fundamental approximation results related to some finite class of orthogonal polynomials called Romanovski{u2013}Jacobi polynomials and Romanovski{u2013}Bessel polynomials and their fractional version on a semi-infinite interval. In Chapter 1, we give a brief introduction to the infinite and finite classes of classical orthogonal polynomials and some essential properties of the infinite classical orthogonal polynomials. In Chapter 2, we introduce some basic properties of the Romanovski-Jacobi polynomials, the Romanovski-Jacobi-Gauss-type quadrature formulae and the associated interpolation, discrete transforms,spectral differentiation and integration techniques in the physical and frequency spaces, and basic approximation results for the weighted projection operator in the nonuniformly weighted Sobolev space. We discuss the relationship between such kind of finite orthogonal polynomials and other classes of infinite orthogonal polynomials.Moreover, we derive spectral Galerkin schemes based on a Romanovski-Jacobi expansion in space and time to solve the Cauchy problem for a scalar linear hyperbolic equation in one and two space dimensions posed in the positive real line