Construction of wavelet sets with certain self-similarity properties
Let \( A \in{\mathbb Z}^{n\times n}\) be a given expansive matrix. A measurable set \(Q \in {\mathbb R}^n\) is called \(A\)-dilation wavelet set if the system \[ \{ |\det A|^{j/2} \, \psi ( A^jX - k) : j \in {\mathbb Z} , \; k \in {\mathbb Z}^n \} \] with \( \widehat \psi = \chi_Q \) is an orthonormal basis for \( {\mathbb R}^n\). In this paper, a class of compact wavelet sets is constructed by using self-affine tiles in the Fourier domain. The new wavelet set do not contain the origin and can be seen as finite unions of integer translates of an integral self-affine tile associated with the transposed matrix \(A^T\). As an application, a counterexample is provided to the conjecture that in dimensions larger than one no dyadic wavelet can be expressed as a finite union of polygons.
scientific article; zbMATH DE number 2144343
scientific article