2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/118036The Littlewood-Paley theory is extended to weighted spaces of distributions on $[-1,1]$ with Jacobi weights $ \w(t)=(1-t)^α(1+t)^β. $ Almost exponentially localized polynomial elements (needlets) $\{ϕ_ξ\}$, $\{ψ_ξ\}$ are constructed and, in complete analogy with the classical case on $\RR^n$, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $\{\ip{f,ϕ_ξ}\}$ in respective sequence spaces.34 pagesClassical Analysis and ODEs42A38, 42B08, 42B15text