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

**A 2006 Preprint by
G. Kyriazis,
P. Petrushev, and
Y. Xu
**

- 2006:11
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) $\{\varphi _ \xi\},\{\psi _ \xi\}$ are constructed and, in complete analogy with the classical case on $R^n$, 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.