On the strong unicity of best Chebyshev approximation of differentiable functions

Abstract:

Let $X$ be a normed linear space, ${U_n}$ an $n$-dimensional Chebyshev subspace of $X$. For $f \in X$ denote by $p(f) \in {U_n}$ its best approximation in ${U_n}$. The problem of strong unicity consists in estimating how fast the nearly best approximants $g \in {U_n}$ satisfying $\left \| {f - g} \right \| \leqslant \left \| {f - p(f)} \right \| + \delta$ approach $p(f)$ as $\delta \to 0$. In the present note we study this problem in the case when $X = {C^r}[a,b]$ is the space of $r$-times continuously differentiable functions endowed with the supremum norm $(1 \leqslant r \leqslant \infty )$.