A Definition of Two-Dimensional Schoenberg Type Operators

Camelia Liliana Moldovan,Radu Păltănea

doi:10.3390/sym12081364

Abstract

In this paper, a way to build two-dimensional Schoenberg type operators with arbitrary knots or with equidistant knots, respectively, is presented. The order of approximation reached by these operators was studied by obtaining a Voronovskaja type asymptotic theorem and using estimates in terms of second-order moduli of continuity.

Highlights

The Schoenberg operators provide a concrete method for obtaining spline approximations of functions
The study of the positive linear Schoenberg operators was the subject of several recent papers, among which we mention here Beutel, Gonska, Kacso and Tachev [1], Tachev [2,3], and Tachev and Zapryanova [4]
Two-dimensional Schoenberg type operators, with h = k = 3 and m = n, with equidistant knots is denoted by Sn,3 : n −1 n −1 (Sn,3 φ)(α, β) =

Summary

Introduction

The Schoenberg operators provide a concrete method for obtaining spline approximations of functions. The subject of multivariate splines was approached by different methods and from various points of view, such as in the papers written by: Curry and Schoenberg [6], Goodman and Lee [7], de Boor and Hollig [8], Karlin et al [9], Goodman [10], Chui [11], Schumaker [12], Conti and Morandi [13], Ugarte et al [14], and Groselj and Knez [15]. Curry and Schoenberg indicated in [6] that the multivariate spline functions can be constructed from volumes of slices of polyhedra; papers can be found that were written toward that direction This idea led to the recurrence relations for multivariate splines presented by Karlin et al in [9]. B-splines is treated in several papers, for example in [12]

Two-Dimensional Schoenberg Type Operators on Arbitrary Nodes

Two-Dimensional Schoenberg Type Operators with Equidistant Knots

Conclusions