Characterising pairs of refinable splines
In a recent paper by \textit{W. Lawton, S. L. Lee}, and \textit{Z. Shen} [Adv. Comput. Math. 3, No. 1-2, 137-145 (1995; Zbl 0828.41006)] a characterization of refinable spline functions with compact support was presented, and it was shown that the integer translates of such a function can only form a Riesz basis if and only if this function is a uniform spline. This paper generalizes the above concept to pairs of spline functions \(\phi_1, \phi_2\) which are refinable in a vector sense, i.e., \(\phi=(\phi_1, \phi_2)^T\) is said to be \(m\)-refinable if \(\phi\) satisfies \[ \phi(x) = \sum_{j=-N}^{N} A(j) \phi(mx-j) \qquad (x \in {\mathbb{R}}) \] with \(2 \times 2\) coefficient matrices \(A(j)\). It is shown that refinability forces a special form of the Fourier transform of \(\phi_1, \phi_2\). Again an explicit characterization of \(\phi_1, \phi_2\) whose integer translates form a Riesz basis can be verified. The B-splines with double integer knots can be found as a special case of that characterization.
scientific article; zbMATH DE number 1121282