The Gibbs–Wilbraham phenomenon in the approximation of |x| by using Lagrange interpolation on the Chebyshev–Lobatto nodal systems

• Published in: Journal of computational and applied mathematics Vol. 414; p. 114403
• Main Authors: Berriochoa, E., Cachafeiro, A., García-Amor, J., García Rábade, H.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.11.2022
• Subjects: Absolute value approximation; Chebyshev nodal systems; Chebyshev–Lobatto nodal systems; Gibbs–Wilbraham phenomena; Lagrange interpolation; Rate of convergence; Absolute value approximation; Gibbs–Wilbraham phenomena; Chebyshev–Lobatto nodal systems; Lagrange interpolation; Chebyshev nodal systems; 65D05; Rate of convergence; 41A05; 42C05
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/j.cam.2022.114403

Along this study we find and deeply analyze a new Gibbs phenomenon. As far as we know, this type of behavior, in different contexts, is connected with functions having jump discontinuities. In our case it is related to the behavior of the Lagrange interpolation polynomials of the continuous absolute value function. Our study is related to the error of the Lagrange polynomial interpolants of the function |x| on [−1,1] taking as nodal system the m+2 nodes of the extended Chebyshev polynomial of the second kind, obtaining that the error behaves like a function of order O(1/m). A detailed description and approximation of the function is presented.