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Regularity of irregular subdivision. (English) Zbl 0957.42022
The paper is concerned with the irregular case of a univariate subdivision scheme; here not only the original samples are nonequally spaced, but also the new grid points introduced by the subdivision scheme need not to be in the middle between the old ones.
The aim of the paper is to study the regularity of limit functions of such irregular subdivision.
It is shown that the commutation formula still holds in the irregular case. For the generalization of the four-point scheme introduced by S. Dubuc [J. Math. Anal. Appl. 114, 185-204, (1986; Zbl 0615.65005)] and by N. Dyn, J. A. Gregory and D. Levin [Constructive Approximation 7, No. 2, 127-147 (1991; Zbl 0724.41011)], the authors show that under mild geometric restrictions the limit function is always \(C^1\), and under slightly stronger restrictions, the limit function is even almost \(C^2\).

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A05 Interpolation in approximation theory
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
26A16 Lipschitz (Hölder) classes
65D05 Numerical interpolation
65T60 Numerical methods for wavelets
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