Some smooth compactly supported tight wavelet frames with vanishing moments (Q305685)

The authors consider wavelet frames, that is functions \(\psi_{\ell, j,\mathbf k}(\mathbf x)\) of the form \[ \psi_{\ell, j,\mathbf k}(\mathbf x)=|\det(A)|^{\frac j2}\psi_\ell(A^j\mathbf x+ \mathbf k), \] where \(\ell=1,\ldots,N\), \(j\in \mathbb Z\), \(\mathbf k\in \mathbb Z^d\), and \(\mathbf x \in \mathbb Z^d\) that form a tight frame in \(L^2(\mathbb R^d)\). The eigenvalues of the dilation matrix \(A\in \mathbb Z^{d\times d}\) are assumed to be greater that \(1\) in absolute value and in most cases it is assumed that \(|\det(A)|=2\). These wavelets are said to have vanishing moments of order \(m\) if the Fourier transforms \(\widehat{\psi_\ell}\) have a zero of order \(m\) at the origin. If the functions \(\psi_\ell\) are linearly independent, then \(N\) is said to be the number of generators of the system. The authors construct families of wavelets functions with one, two or \(2^d+d\) generators, the last construction giving a family having any desired degree of regularity. For dimension \(2\) this construction is simplified in the last sections. None of these constructions rely on tensor products of lower-dimensional functions but do rely on extensions of results for dyadic dilations to general dilation matrices \(A\) as well as e.g.\ the so called Oblique Extension Principle.