Criteria for hierachical bases in Sobolev spaces
The object of this paper is to determine, both theoretically and numerically, the range of \(s\) for which a hierarchical system \[ \Psi:= \{\Phi(x- \alpha),\;2^{jd/2} \Psi^\lambda(2^jx- \alpha): \alpha\in \mathbb{Z}^d,\;j= 0,1,2,\dots,\;\lambda= 1,\dots, 2^d- 1\} \] forms a Riesz basis for the Sobolev space \(H^s(\mathbb{R}^d)\). This system is derived by a dyadic multiresolution analysis \[ V_0\subset V_1\subset\cdots\subset V_j\subset\cdots \] of closed subspaces of \(L_2(\mathbb{R}^d)\) with scaling function \(\phi\in V_0\) of compact support by specifying \(2^d- 1\) functions \(\Psi^\lambda\in V_1\), \(\lambda= 1,\dots, 2^d\), of compact support. This involves determining the smoothness of the dual system. The elements of the dual system consist of noncompactly supported functions. The authors show how to determine the exact range of Sobolev exponents \(s\) in the multivariant case using spectral properties of transfer operators. For Stevenson's hierarchical basis they find the estimation \(-0,990236\dots< s< 3/2\).
scientific article; zbMATH DE number 1415638
scientific article