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.