Whitney's and Seeley's type of extensions for maps defined on some Banach spaces

TL;DR

This paper generalizes Whitney's and Seeley's extension theorems for maps with bounded derivatives on segments in certain Banach spaces, enabling extensions with preserved smoothness, with implications for function approximation and manifold learning.

Contribution

It extends Whitney's and Seeley's theorems to broader classes of Banach spaces, providing new tools for smooth function extension in infinite-dimensional settings.

Findings

Extension theorems for maps with bounded derivatives in Banach spaces

Generalizations of Whitney's and Seeley's theorems

Applications to function approximation and manifold learning

Abstract

Let $X=C[0,1]$, and $Y$ be an arbitrary Banach space. Consider a collection of open segments $\{V_i \}\subset X$. Suppose the map $f: \cup_i V_i \to Y$ has $q$ bounded Fr\'echet derivatives ($q=0,1,...,\infty$), and $f$ and all its derivatives have continuous bounded limits at the boundary. Then, subject to some non-intercept condition for the segments $V_i$, the map $f$ can be extended to $F: X\to Y$, so that $F_{|\,\cup_i V_i}=f$ and $F$ has $q$ bounded derivatives. We prove similar Whitney's Extension theorem generalizations for some other Banach spaces. We also prove Seeley Extension theorem for $X=C[0,1].$ These results are related to the problems of function approximation, and manifold learning, which are of central importance to many applied fields.