Two-scale homogeneous functions in wavelet analysis.
In this paper two-scale homogeneous functions, satisfying \(h(x) = \lambda \cdot h(2x)\) for some constant \(\lambda\), are considered. Let \(\phi = (\phi_1, \ldots , \phi_r)^T\) be a refinable vector of functions, i.e., \[ \phi (x) = \sum_{k\in {\mathbb Z}} a(k) \phi (2x-k), \] where \(a := (a(k))_{k\in {\mathbb Z}}\) is a sequence of \(r \times r\) matrices. The author investigates two-scale homogeneous functions that are contained in the shift invariant space \(S(\phi)\) generated by the integer translates of \(\phi_1, \ldots , \phi_r.\) This study leads to some interesting conclusions. There is a one-to-one correspondence between two-scale homogeneous functions of order \(\lambda \) and left eigenvectors of \((a(2j-k))^{N-1}_{j,k=0}\) to the eigenvalue \(\lambda\). It can now be shown that among all refinable function vectors with fixed support the B-spline vector with multiple knots yields optimal accuracy. Further, the subdivision operator related to \((a(k))_{k \in {\mathbb Z}}\) is shown to have no eigenvalues in \(l_p({\mathbb Z})\). Finally, the reconstruction of \(\phi\) from two-scale homogeneous functions is considered.
scientific article; zbMATH DE number 1921137