  The affine synthesis operator is shown to map the coefficient space $\ell^p$
surjectively onto $L^p$, for $0 < p \leq 1$. Here the synthesizer need satisfy
only mild restrictions, for example having nonzero integral or else
periodization that is real-valued, nontrivial and bounded below.
  Consequences include an affine atomic decomposition of $L^p$.
  Tools include an analysis operator that acts nonlinearly, in contrast to the
usual linear analysis operator for $p>1$.
