### Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces

The Littlewood-Paley theory is extended to weighted spaces of distributions on [-1,1] with Jacobi weights $w\left(t\right)={(1-t)}^{\alpha}{(1+t)}^{\beta}$. Almost exponentially localized polynomial elements (needlets) ${\phi}_{\xi}$, ${\psi}_{\xi}$ are constructed and, in complete analogy with the classical case on ℝⁿ, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $\u27e8f,{\phi}_{\xi}\u27e9$ in respective sequence spaces.