\(M\)-band scaling function with filter having vanishing moments two and minimal length
In this paper, the authors consider the Hölder continuity, local linearity, linear independence and interpolation problem for the \(M\)-band scaling functions with two vanishing moments and minimal length. Thus the \(M\)-band filter transfer function is factorized as \(H(z)=({{1-z^M}\over {M(1-z)}})^2(\alpha+\beta z)z^k\). The scaling function \(\phi_M\) is obtained from the usual refinement equation \({\widehat{\phi}}(z)=H({{z}\over{M}}) {\widehat{\phi}}({{z}\over{M}})\). In particular, the authors obtain that \(\phi_M\) is locally linear on an open set with full measure and locally dependent when \(M\geq 3\); \(\widetilde{\phi}_M\) is differentiable at \(M\)-adic points when \(M=3\), and \(\widetilde{\phi}_M\) is not interpolatable at \(Z\) when \(M=11\), where \(\widetilde{\phi}_M(x) = \phi(2+1/(M-1)-x)\).
scientific article; zbMATH DE number 1217011