Jacobi decomposition of weighted Triebel–Lizorkin and Besov spaces
Volume 186 / 2008
Studia Mathematica 186 (2008), 161-202
MSC: 42A38, 42B08, 42B15.
DOI: 10.4064/sm186-2-3
Abstract
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 $\mathbb R^n$, 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.
