OnTheOrthogonal Rational Functions

Abstract

Formorethanacentury,orthogonalpolynomials(OP)havebeenacornerstoneoffunc tional and numerical analysis. They play a fundamental role in approximation, interpola tion, quadrature,andthenumericalsolutionofintegralequations.Theirwell-established theoretical framework has enabled the development of many efficient computational me thods. However, their rigid structure sometimes limits their efficiency, especially when dealing with integrals involving singularities or irregular behaviors. Orthogonal rational functions (ORF) then emerge as a natural and more flexible extension of OPs. By introducing adjustable poles, they preserve orthogonality while providing better adaptability to the characteristics of the problem under study. They thus open new perspectives for modern scientific computing, particularly in the solution of integral equations. This thesis is organized around two main complementary directions : 1. The theoretical and analytical construction of ORFs in various functional spaces (Hilbert, weighted Lebesgue, Hardy), and the development of rational quadrature formulas of the Gauss–Chebyshev type; 2. Their application to the numerical solution of Fredholm and Volterra integral equations through projection and collocation methods, accompanied by a detailed analysis of convergence, stability, and numerical performance. The ORFs are used as an adaptive functional basis to approximate the solutions of integral equations. The results demonstrate a significant improvement in accuracy, stability, and convergence rate compared with classical approaches based on orthogonal polynomials. This work thus establishes a clear link between theory and numerical computation, confirming the potential of ORFs as efficient tools for rational approximation and the numerical solution of complex integral equations.