2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/125510The 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 $p>1$.29 pagesClassical Analysis and ODEs41A30; 46E30text