Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces

Abstract

The Littlewood-Paley theory is extended to weighted spaces of distributions on $[-1,1]$ with Jacobi weights $ \w(t)=(1-t)^α(1+t)^β. $ Almost exponentially localized polynomial elements (needlets) $\{{\varphi }_{\xi }\}$, $\{{\psi }_{\xi }\}$ 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 $\{\ip{f,ϕ_ξ}\}$ in respective sequence spaces.

34 pages

Cited by in corpus (1)

• Decomposition of Triebel-Lizorkin and Besov spaces in the context of Laguerre expansions