Refined error estimates and Fourier techniques in the approximation of higher-order derivatives via Gegenbauer orthogonal expansions

Abstract

This paper investigates the approximation of higher-order derivatives using integral operators constructed from Gegenbauer polynomials. We establish key equivalence results connecting the approximation error in the L²-norm to the decay of high-frequency components in the Fourier domain. A notable contribution is the derivation of Titchmarsh-type theorems that relate the solid average and Lanczos operators to the smoothness of the function being approximated. Additionally, we derive conditions under which the Fourier transform of functions belongs to \(L^\beta(\mathbb{R})\), depending on the smoothness of the function and the parameters of the approximation.