Simple regularity criteria for subdivision schemes. (English) Zbl 0761.42016

Summary: Convergent subdivision schemes arise in several fields of applied mathematics (computer-aided geometric design, fractals, compactly supported wavelets) and signal processing (multiresolution decomposition, filter banks). In this paper, a polynomial description is used to study the existence and Hölder regularity of limit functions of binary subdivision schemes. Sharp regularity estimates are derived; they are optimal in most cases. They can easily be implemented on a computer, and simulations show that the exact regularity order is accurately determined after a few iterations. Connection is made to regularity estimates of solutions to two-scale difference equations as derived by Daubechies and Lagarias, and other known Fourier-based estimates. The former are often optimal, while the latter are optimal only for a subclass of symmetric limit functions.


42C15 General harmonic expansions, frames
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
39A10 Additive difference equations
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