Today I read a paper titled “Complex Eigenvalues for Binary Subdivision Schemes”
The abstract is:
Convergence properties of binary stationary subdivision schemes for curves have been analyzed using the techniques of z-transforms and eigenanalysis
Eigenanalysis provides a way to determine derivative continuity at specific points based on the eigenvalues of a finite matrix
None of the well-known subdivision schemes for curves have complex eigenvalues
We prove when a convergent scheme with palindromic mask can have complex eigenvalues and that a lower limit for the size of the mask exists in this case
We find a scheme with complex eigenvalues achieving this lower bound
Furthermore we investigate this scheme numerically and explain from a geometric viewpoint why such a scheme has not yet been used in computer-aided geometric design