Affine synthesis onto $L^p$ when $0<p\le 1$

Abstract

The affine synthesis operator is shown to map the coefficient space ${\ell }^p$ surjectively onto $L^p$, for $0<p\le 1$. 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 $p>1$.

29 pages