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. 
Huy-Qui Bui and Richard S. Laugesen