Stability and Convergence of Extension Schemes to Continuous Functions in General Metric Spaces
Erwan Le Gruyer, J. C. Archer · SIAM Journal on Mathematical Analysis · 1996
For any E, $E'$ general metric spaces, we formulate the concept of stability of an extension scheme $\mathcal{E}$ ($\varphi $ continuous mapping from some closed subset of E into $E'$, $\mathcal{E}(\varphi )$ continuous and extending $\varphi $). We show that, when $E' = \mathbb{R}$, stable extension schemes always exist and that the classical extension schemes in the literature are instable. We also show that, when $E'$ is complete, any stable extrapolation scheme $\mathcal{E}$ ($\varphi $ mapping from some discrete and closed subset of E into $E'$, $\mathcal{E}(\varphi )$ continuous and extending $\varphi $) has a unique extension to a stable extension scheme: this result establishes a link between the problem of extrapolation, which usually refers to numerical analysis, and the problem of extension, which also concerns pure mathematics.