The Littlewood-Paley theory is extended to weighted spaces of distributions on [-1,1] with Jacobi weights $w\left(t\right)={(1-t)}^{\alpha}{(1+t)}^{\beta}$. Almost exponentially localized polynomial elements (needlets) ${\phi}_{\xi}$, ${\psi}_{\xi}$ are constructed and, in complete analogy with the classical case on ℝⁿ, it is shown that weighted Triebel-Lizorkin and Besov spaces can be characterized by the size of the needlet coefficients $\u27e8f,{\phi}_{\xi}\u27e9$ in respective sequence spaces.

Our primary goal in this preamble is to highlight the best of Vasil Popov’s
mathematical achievements and ideas. V. Popov showed his extraordinary talent
for mathematics in his early papers in the (typically Bulgarian) area of approximation
in the Hausdorff metric. His results in this area are very well presented
in the monograph of his advisor Bl. Sendov, “Hausdorff Approximation”.

* This work has been supported by the Office of Naval Research Contract Nr. N0014-91-J1343,
the Army Research Office Contract Nr. DAAD 19-02-1-0028, the National Science Foundation
grants DMS-0221642 and DMS-0200665, the Deutsche Forschungsgemeinschaft grant SFB 401,
the IHP Network “Breaking Complexity” funded by the European Commission and the Alexan-
der von Humboldt Foundation.
Adaptive Finite Element Methods (AFEM) are numerical procedures
that approximate the solution to a partial...

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