Quarkonial frames with compression properties
Let \(\sum_{k\in Z} \phi (x-k) =1\), \(x\in\mathbb R\), be a partition of unity in \(\mathbb R\), where \(\phi\) is a smooth compactly supported function. Let \(\phi_p (x) = x^p \phi (x)\), \(p\in \mathbb N_0\). The paper deals with the quarks or quarklets \[ \phi_{p,j,k} (x) = 2^{j/2} \phi_p \big( 2^j x-k), \qquad p,j \in \mathbb N_0, \quad k\in \mathbb Z, \quad x\in \mathbb R. \] These basic functions are highly redundant compared with the usual wavelet bases. But they are easy to construct and highly flexible. In addition, they are frames in \(L_2 ( \mathbb R)\) and, in particular, in the (fractional) Sobolev spaces \(H^s ( \mathbb R)\), \(s>0\). Section 3 deals with related estimates of Jackson and Bernstein type. The frame property is studied in Section 4 (frames in \(L_2 ( \mathbb R)\)) and Section 5 (frames in \(H^s ( \mathbb R)\), \(s>0\)). This is applied in Section 6 to some elliptic problems.
scientific article; zbMATH DE number 6798984