Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Affine synthesis and coefficient norms for Lebesgue, Hardy and Sobolev
  spaces   The affine synthesis operator is shown to map the mixed-norm sequence space
$\ell^1(\ell^p)$ surjectively onto $L^p(\Rd), 1 \leq p < \infty$, assuming the
Fourier transform of the synthesizer does not vanish at the origin and the
synthesizer has some decay near infinity. Hence the standard norm on $f \in
L^p(\Rd)$ is equivalent to the minimal coefficient norm of realizations of $f$
in terms of the affine system.
  We further show the synthesis operator maps a discrete Hardy space onto
$H^1(\Rd)$, which yields a norm equivalence for Hardy space involving
convolution with a discrete Riesz kernel sequence.
  Coefficient norm equivalences are established also for Sobolev spaces, by
applying difference operators to the coefficient sequences.

10
The categories of this paper are: q-bio
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
25
The categories of this paper are: patt-sol
29
The categories of this paper are: funct-an
15
The categories of this paper are: nucl-ex
16
The categories of this paper are: stat
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
3
The categories of this paper are: physics
33
The categories of this paper are: plasm-ph
31
The categories of this paper are: ao-sci
22
The categories of this paper are: chao-dyn