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

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 Fréchet 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.