Stability Analysis of Grünwald Interpolation Operators on Chebyshev Nodes
Vinaya P C
TL;DR
The paper extends Grünwald's convergence results for interpolation on Chebyshev nodes by analyzing averaged Lagrange operators G_n when the interpolation nodes are perturbed away from Chebyshev grids. It establishes uniform convergence for a broad class of perturbed and uniformly distributed nodal sets, derives a Voronovskaja-type estimate with a suboptimal n^{−1/3} rate, and provides explicit modulus-of-continuity bounds to quantify convergence. The work identifies conditions under which the Grünwald operators stabilize and delivers concrete, operator-norm and pointwise error controls relevant for approximation theory. This broadens the applicability of Grünwald-type interpolation by linking node distribution to convergence stability and rate estimates.
Abstract
In 1941, G. Grünwald proved the convergence of a sequence of operators constructed using classical Lagrange interpolation at Chebyshev nodes. In this work, we establish a perturbed version of Grünwald's result, thereby extending the class of admissible nodal points. Specifically, we provide sufficient conditions for convergence when the interpolation nodes are of the form {cos eta_k} for k = 1, ..., n, where {eta_k} is a general sequence. We refer to these operators as Grünwald operators. In particular, we prove a convergence result when {eta_k} is equidistant and uniformly distributed. We establish a Voronovskaja-type estimate for the convergence of these operators and derive quantitative results using modulus of continuity.