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, italic $ and to the unit ball $B^d$ in $R^d$ with weights $W_\mu(x)= (1-|x|^2)^{\mu-1/2}$, $\mu \ge 0$. Almost exponentially localized polynomial elements (needlets) $\{\varphi_\xi\}$, $\{\psi_\xi\}$ are constructed and, in complete analogy with the classical case on $R^d$, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $\{\langle f,\varphi_\xi\rangle\}$ in respective sequence spaces.