2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167011We introduce a harmonic analysis for a class of affine iteration models in $\br^d$. Using Hilbert-space geometry, we develop a new duality notion for affine and contractive iterated function systems (IFSs) and we construct some identities for the Fourier transform of the measure corresponding to infinite Bernoulli convolutions.new version, we followed the comments of the referee and rewrote the introductory sectionsDynamical Systems42A85, 42C25, 26A30, 28A80, 60G42, 47B80text