Compactly supported tight affine frames with integer dilations and maximum vanishing moments
Let \(\phi\) be a compactly supported \(M\)-dilation scaling function associated with a finite real-valued scaling sequence. Let \(f_{j,k}(x) = M^{j/2}f(M^jx-k)\), where \(j,k \in {\mathbb Z}\), and integer \(M \geq 2\). The paper under review studies tight frames for \(L^2({\mathbb R})\) generated by a family \(\{\psi^l_{j,k}: l=1, \ldots\), \(N, j,k \in {\mathbb Z}\}\), where the functions \(\psi^l\) are defined by the scaling equations: \[ \hat \psi^l (\omega) = Q_l(e^{-i\omega/M})\widehat \phi(\omega/M), \] where \(Q_l\), \(l=1, \ldots, N\), are real Laurent polynomials vanishing at 1, in the form of \[ Q_l(z) = (1-z)^{L_l}q_l(z), \] \(q_l(1) \neq 0\). In this formulation \(L_l\) is the order of vanishing moments for \(\psi^l\). The authors characterize such tight frames through the existence of a so-called \textit{vanishing moments recovery (VMR) function}. The properties of VMR functions are studied and used to provide explicit constructions of tight wavelet frames with additional vanishing moments.
scientific article; zbMATH DE number 1896701