  The Littlewood-Paley theory is extended to weighted spaces of distributions
on $[-1,1]$ with Jacobi weights $ \w(t)=(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 $\RR^n$, it is shown that weighted Triebel-Lizorkin and Besov spaces can be
characterized by the size of the needlet coefficients $\{\ip{f,\phi_\xi}\}$ in
respective sequence spaces.
