2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/115614The 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.Added references, and improved several proofs. Also added Appendix C, which connects the paper to Banach frame theoryClassical Analysis and ODEs41A30; 42B30; 46E35text