Stability Analysis of Gr\"unwald Interpolation Operators on Chebyshev Nodes

Abstract

In 1941, G. Gr\"unwald 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 Gr\"unwald'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 Gr\"unwald 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.