Interpolatory blending net subdivision schemes of Dubuc-Deslauriers type
The authors present a general construction of families of interpolatory net subdivision schemes generating limit bivariate functions by repeated refinements of nets of univariate functions. In this construction, the refined nets are obtained by sampling on refined grids Gordon blending interpolation to the coarser nets. For a given scheme, the blending function in the Gordon interpolants is the same at all refinement levels, and has properties which relates it to a corresponding Dubuc-Deslauriers interpolatory scheme [\textit{G. Deslauriers} and \textit{S. Dubuc}, Constructive Approximation 5, No. 1, 49--68 (1989; Zbl 0659.65004)] refining points. The main results in this paper show that these schemes converge to limit functions having the same integer smoothness as the limits of the corresponding Dubuc-Deslauriers schemes using the properties of the blending functions together with the general analysis tools by \textit{C. Conti} and \textit{N. Dyn} [J. Comput. Appl. Math. 236, No. 4, 461--475 (2011; Zbl 1235.65020)]. These results are proved for net subdivision schemes corresponding to the first 84 members of the Dubuc-Deslauriers family, and conjectured for the rest. A concrete example of a family of piecewise polynomial blending functions is considered, together with the corresponding family of net subdivision schemes.
Dubuc-Deslauriers interpolatory subdivision
scientific article; zbMATH DE number 6112271