Convergence of irregular Hermite subdivision schemes
The authors consider Hermite subdivision schemes on the real line in which the initial function data, consisting of the function value and its first derivative, are specified at a collection of sample points. However, rather than assuming that the sample points on the real line are regularly spaced for each level of subdivision, the authors allow for irregular spacing. For such schemes, they construct an associated scalar subdivision scheme, and show that for a quasi-regular grid, and under some minor assumptions about the scheme, the convergence of the associated scalar subdivision scheme implies the convergence of the Hermite subdivision scheme whenever the function values of the initial data form a convergent sequence. The authors then go on to consider \((2N+2)\)-point interpolatory Hermite subdivision schemes: a specific family of schemes defined on a grid that is a perturbation of a regular grid. In particular they show that these schemes converge if they converge on a regular grid. The article concludes with a brief discussion of how some scalar uniform convergence results may be extended to \(L_p\) convergence. Although not entirely self-contained, the article is intended for a general audience versed in concepts from real analysis; however, readers will find a prior knowledge of subdivision schemes useful. While organized, the article does contain occasional typographical errors.
scientific article; zbMATH DE number 5869115