Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Convex $C^1$ extensions of $1$-jets from compact subsets of Hilbert
  spaces   Let $X$ denote a Hilbert space. Given a compact subset $K$ of $X$ and two
continuous functions $f:K\to\mathbb{R}$, $G:K\to X$, we show that a necessary
and sufficient condition for the existence of a convex function $F\in C^1(X)$
such that $F=f$ on $K$ and $\nabla F=G$ on $K$ is that the $1$-jet $(f, G)$
satisfies (1) $f(x)\geq f(y)+ \langle G(y), x-y\rangle$ for all $x, y\in K$,
and (2) if $x, y\in K$ and $f(x)= f(y)+ \langle G(y), x-y\rangle$ then
$G(x)=G(y)$. We also solve a similar problem for $K$ replaced with an arbitrary
bounded subset of $X$, and for $C^1(X)$ replaced with the class
$C^{1,u}_{b}(X)$ of differentiable functions with uniformly continuous
derivatives on bounded subsets of $X$.

20
The categories of this paper are: acc-phys
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
19
The categories of this paper are: eess
18
The categories of this paper are: econ
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
23
The categories of this paper are: solv-int
8
The categories of this paper are: hep-ex