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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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