How to extend a continuous function from domain open in $H^k$ to a domain open in $R^k$?

Question

Assume $f:U\mapsto V$ is continuous and of class $C^r$, and $U$ open in $H^k$ but not $R^k$. How do I extend $f$ to $g:U'\mapsto V$, $g$ is also continuous and of class $C^r$, $U'$ is open in $R^k$, and $g$ agrees with $f$ on $U$?

Answer 1

There is a general Whitney extension theorem for extension of smooth functions from closed subsets of $\mathbb R^k$. But in your case a higher order reflection will do the job, and in a quite explicit way.It is the same for $k=1$ as for general $k$, so I'll state it in one dimension: $$f(-x) = \sum_{j=0}^r a_j f(2^{-j}x) \tag1$$ where the coefficients $a_j$ are chosen so that for $m=0,\dots,r$ we have $$(-1)^m = \sum_{j=0}^r a_j 2^{-jm} \tag2$$ The meaning of (2) is that every polynomial of degree at most $r$ is preserved by the reflection formula. Writing down the Taylor approximation to $f$ of order $r$, centered at a point of the reflection plane, you will see that the reflected function also satisfies it.