APPROXIMATION BY ORTHOGONAL POLYNOMIALS

Abstract

This thesis investigates the theory and applications of orthogonal polynomials in approximation theory, integrating foundational analysis with modern computational methods. Beginning with the formulation of orthogonality in weighted inner product spaces and the derivation of the three-term recurrence relation, the study examines classical families such as Legendre, Chebyshev, Hermite, and Laguerre polynomials and their roles in minimizing approximation error in L2 and L∞ norms. The convergence behavior of orthogonal series is analyzed through Jackson-type estimates, Lebesgue constants, and asymptotic decay of coefficients, highlighting the influence of function smoothness and phenomena such as Gibbs oscillations. Computational aspects, including FFT-based coefficient evaluation and Gaussian quadrature, are connected to spectral methods for differential equations, demonstrating exponential convergence for smooth solutions. The work further extends to non classical weights and Sobolev orthogonal polynomials, emphasizing their relevance in variational formulations and energy norm approximations.