When do perturbed Chebyshev--Lobatto points remain Chebyshev?

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 $x_j=\cos (j\pi \mathrm{/}n+{\epsilon }_j)$, with angular perturbations $\mathrm{\mid }{\epsilon }_j\mathrm{\mid }\le {\sigma }_n$. The study is motivated by numerical experiments showing a broad stable region when the mesh fraction $n{\sigma }_n$ is small and rapid amplification for larger perturbations; the observed transition region is consistent with the curve $n{\sigma }_n\asymp (\log n{)}^{-1}$. The main result is a deterministic worst-case stability estimate: if $n{\sigma }_n(\log n+1)$ 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 $1\mathrm{/}n$ 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.