## Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces.(English)Zbl 1283.42011

Summary: 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 $$\{ip{f,\varphi_\xi}\}$$ in respective sequence spaces.

### MSC:

 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42B08 Summability in several variables 42B15 Multipliers for harmonic analysis in several variables
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