Factorization theorems for univariate splines on regular grids
A systematic analysis of the possible factorizations of univariate splines is presented. The author uses some of the techniques known in the multivariate situation together with some new ones to obtain factorization results for univariate splines. These results are used for characterizations of the splines of minimal support and singular exponential B-splines. The main factorization theorem can be stated as follows: Let \(\phi\) be a given univariate compactly supported distribution. Denote by S(\(\phi\)) the space spanned by its integer translates, by H(\(\phi\)) the subspace of all exponentials in S(\(\phi\)) and by \(K_{\phi}\) the kernel of the associated semi-discrete convolution \(\phi^*\). Then \(\phi\) can be expressed as \(\phi =\rho (\nabla)\tau *M\), where \(\rho\) (\(\nabla)\) is a finite difference operator, \(\tau\) is a distribution of smaller support satisfying \(H(\tau)=K_{\tau}=\{0\}\), and M is a spline which depends on H(\(\phi\)) but not on \(\phi\) itself.
scientific article; zbMATH DE number 4208738