On the filtered polynomial interpolation at Chebyshev nodes

TL;DR

This paper investigates a filtered polynomial interpolation method at Chebyshev nodes that ensures uniform convergence and near-best approximation, with conditions for bounded Lebesgue constants and reduced Gibbs phenomenon, supported by numerical tests.

Contribution

It provides necessary and sufficient conditions for bounded Lebesgue constants in filtered interpolation, improving convergence and approximation quality over classical methods.

Findings

Lebesgue constants are bounded under simple inequalities on Jacobi weight exponents.

Filtered interpolants achieve near-best approximation errors that tend to zero as nodes increase.

Numerical experiments confirm theoretical results and demonstrate reduced Gibbs phenomenon.

Abstract

The paper deals with a special filtered approximation method, which originates interpolation polynomials at Chebyshev zeros by using de la Vall\'ee Poussin filters. These polynomials can be an useful device for many theoretical and applicative problems since they combine the advantages of the classical Lagrange interpolation, with the uniform convergence in spaces of locally continuous functions equipped with suitable, Jacobi--weighted, uniform norms. The uniform boundedness of the related Lebesgue constants, which equals to the uniform convergence and is missing from Lagrange interpolation, has been already proved in literature under different, but only sufficient, assumptions. Here, we state the necessary and sufficient conditions to get it. These conditions are easy to check since they are simple inequalities on the exponents of the Jacobi weight defining the norm. Moreover, they are necessary and sufficient to get filtered interpolating polynomials with a near best approximation error, which tends to zero as the number $n$ of nodes tends to infinity. In addition, the convergence rate is comparable with the error of best polynomial approximation of degree $n$, hence the approximation order improves with the smoothness of the sought function. Several numerical experiments are given in order to test the theoretical results, to make a comparison with the Lagrange interpolation at the same nodes and to show how the Gibbs phenomenon can be strongly reduced.