Rioul, Olivier Simple regularity criteria for subdivision schemes. (English) Zbl 0761.42016 SIAM J. Math. Anal. 23, No. 6, 1544-1576 (1992). 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. Cited in 64 Documents MSC: 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 Keywords:subdivision algorithms; Sobolev regularity; convergent subdivision schemes; compactly supported wavelets; Hölder regularity; binary subdivision schemes; two-scale difference equations PDFBibTeX XMLCite \textit{O. Rioul}, SIAM J. Math. Anal. 23, No. 6, 1544--1576 (1992; Zbl 0761.42016) Full Text: DOI