Let $a\le b$ and $k\ge 0$ be given and fixed. Let furthermore $x$ and $y$ denote two different elements of a Hilbert space $H$. Suppose $u:\mathbb{R}\rightarrow H$ is a $C^k$-embedding connecting $x$ and $y$, s.t. the derivatives up to order $k$ vanish at infinity and $f:\mathbb{R} \rightarrow \mathbb{R}$ a given $C^k$-map with support in $[a,b].$ Then it it possible to construct a $C^k$map $\tilde{f}:H\rightarrow \mathbb{R}$ satisfying $\tilde{f}(u(s))=f(s)$ for any $s\in \mathbb{R}.$ (This can be done using a tubular neighborhood and a suitable cutoff function.) 


Is it possible to make the map $f\mapsto\tilde{f}$ continuous? The domain of this map should be the space of $C^k$-embeddings (as above) times $\mathbb{R}$-valued $C^k$- functions supported in $[a,b]$. And the codomain should be the space of $C^k$-maps on $H$.



continuity of extension of maps along curves
Some rough ideas to construct a continuous map. I'm not sure that
there aren't any obstructions this might run into.


First, it's nice to have an (explicit) construction of the map
$F:u,f \mapsto \tilde{f}$ to be able to say anything about continuity.


An explicit construction of a tubular neighborhood of $u([a,b])$ can
be made via the exponential mapping of the normal bundle, but this
loses one degree of smoothness. One could use a smoothing operator on
$u([a,b])$ first. Both $u([a,b])$ and
$u(\mathbb{R}\setminus(a-\delta,b+\delta))$ are compact, so these are
separated by a finite distance $\epsilon$ and therefore there should
be a neighborhood of $u$ which still has this property for a finite
distance, say $\epsilon/2$. I think that such compactness arguments
should allow you to proof continuity of the map $F$. You may want to
look into the Omega lemma, which states conditions for the composition
mapping $f,g \mapsto f \circ g$ to be continuous. As a reference, you
may look into Abraham, Marsden, Ratiu ``Manifolds, tensor analysis,
and applications''.


