When do perturbed Chebyshev--Lobatto points remain Chebyshev?

Abstract

Chebyshev points are distinguished in polynomial interpolation by the logarithmic growth of their Lebesgue constants. This paper asks a simple question: how much can Chebyshev points be perturbed before they cease to behave like Chebyshev points? We study perturbed Chebyshev--Lobatto nodes $x_j=\cos (j\ddot{I}€/n+{\epsilon }_j)$, with angular perturbations $|{\epsilon }_j|\le \ddot{I}{ƒ}_n$. The study is motivated by numerical experiments showing a broad stable region when the mesh fraction $n\ddot{I}{ƒ}_n$ is small and rapid amplification for larger perturbations; the observed transition region is consistent with the curve $n\ddot{I}{ƒ}_n\asymp (\log n{)}^{-1}$. The main result is a deterministic worst-case stability estimate: if $n\ddot{I}{ƒ}_n(\log n+1)$ is bounded by a sufficiently small constant, then the Lebesgue constant remains logarithmic. The proof uses the cosine parametrization and Bernstein's inequality for trigonometric polynomials, thereby exploiting the angular geometry of the Chebyshev--Lobatto grid rather than a Markov inequality in the physical variable. We also give a worst-case obstruction at the angular mesh scale, showing that perturbations of order $1/n$ cannot be allowed uniformly. Consequences are derived for analytic interpolation in Bernstein ellipses, for the absence of Runge-type divergence in the stable analytic regime, and for pseudospectral differentiation. Numerical experiments illustrate the transition in the Lebesgue constants, the shape of the associated Lebesgue functions, Runge-function interpolants, and finite-precision differentiation errors.

17 pages, 4 figures