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

TL;DR

This paper extends Gr"unwald's classical convergence results for interpolation operators at Chebyshev nodes to a broader class of nodes, providing new convergence conditions and quantitative estimates.

Contribution

It introduces a perturbed version of Gr"unwald operators, establishing convergence criteria for more general node sequences beyond Chebyshev points.

Findings

Convergence proven for nodes of the form {cos eta_k} with general eta_k sequences.

Established a Voronovskaja-type asymptotic estimate for these operators.

Derived quantitative convergence results using modulus of continuity.

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.