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(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. 
George Kyriazis, Pencho Petrushev and Yuan Xu