``Compactly'' supported frames for spaces of distributions on the ball
The authors study the problem of the construction of multiscale frames on the unit ball \(B^d\) in \(\mathbb R^d\) consisting of the \(C^{\infty}\) functions with small supports which shrink at higher scales. The construction relies on a general scheme for construction of frames from [the authors, J. Funct. Anal. 257, No. 7, 2159--2187 (2009; Zbl 1179.42025)] and frames called ``needlets'' for weighted Triebel-Lizorkin and Besov spaces on \(B^d\) developed by the authors and \textit{Y. Xu} [Proc. Lond. Math. Soc. (3) 97, No. 2, 477--513 (2008; Zbl 1153.41004)] and \textit{P. Petrushev} and \textit{Y. Xu} [Constr. Approx. 27, No. 2, 121--148 (2008; Zbl 1213.42153)]. The authors construct frames that can be used to decompose weighted Triebel-Lizorkin and Besov spaces on \(B^d\) with weight \(w_{\mu}(x) := (1-|x|^2)^{\mu - 1/2}\), \(\mu\) half integer and nonnegative.
scientific article; zbMATH DE number 6018282
scientific article