ON THE DENSE UNBOUNDED DIVERGENCE OF THE DISCRETE BEST APPROXIMATION

Abstract

A classic theorem of Approximation Theory states the uniform convergence of the best approximation polynomials, concerning the Banach space $C$ of all real-valued continuous functions defined on the interval $[-1,1]$ of $\mathbb{R}$, in supremum norm. By contrast, the main result of this paper highlights the phenomenon of double condensation of singularities (meaning unbounded divergence on large subsets of $C$ and $[-1,1]$, in topological sense) for the discrete best approximation on Chebyshev nodes.