Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Vector-valued local approximation spaces   We prove that for every Banach space $Y$, the Besov spaces of functions from
the $n$-dimensional Euclidean space to $Y$ agree with suitable local
approximation spaces with equivalent norms. In addition, we prove that the
Sobolev spaces of type $q$ are continuously embedded in the Besov spaces of the
same type if and only if $Y$ has martingale cotype $q$. We interpret this as an
extension of earlier results of Xu (1998), and Mart\'inez, Torrea and Xu
(2006). These two results combined give the characterization that $Y$ admits an
equivalent norm with modulus of convexity of power type $q$ if and only if
weakly differentiable functions have good local approximations with
polynomials.

22
The categories of this paper are: chao-dyn
24
The categories of this paper are: dg-ga
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
26
The categories of this paper are: bayes-an
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
16
The categories of this paper are: stat
19
The categories of this paper are: eess
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th