On Refinement Equations Determined by Pólya Frequency Sequences
T. N. T. Goodman, Charles A. Micchelli · SIAM Journal on Mathematical Analysis · 1992
The refinement equation \[ \phi (x) = \sum_{i \in \mathbb{Z}} {a_i \phi (2x - i)} ,\quad x \in \mathbb{R}\] for a given sequence ${\bf a} = \{ {a_j :i \in \mathbb{Z}} \}$ has found important application in the study of both Stationary Subdivision Schemes for the generation of curves and surfaces as well as the construction of orthonormal wavelets by means of multiresolution analysis. The main goal here is to study properties of the solution of this equation when the sequence ${\bf a}$ is a Pólya frequency sequence. In the case that ${\operatorname{supp}}{\bf a}: = \{ {k:a_k e 0,k \in \mathbb{Z}} \}$ is finite the refinement equation is also considered when \[ a(z) = \sum_{j = 0}^n {a_j z^j } \] is a Hurwitz polynomial (has all zeros in the left-half plane).