Chebyshev--Schoenberg operators
The present paper generalizes the results of \textit{M.-L. Mazure} [Numer. Algorithms 52, No. 1, 93--128 (2009; Zbl 1200.41023)] to Schoenberg-type operators. It is shown that a given spline space based on a given extended Chebyshev space gives birth to infinitely many variation-diminishing operators of Schoenberg-type, characterized by the two-dimensional Chebyshev spaces they reproduce. It is shown that the set of these two-dimensional spaces is stable under knot insertion and dimension elevation. The author considers the results of this article as originating in the powerful theory of Chebyshevian blossoms. Chebyshev-Schoenberg operators based on a given Chebyshevian spline space are defined. Building Chebyshev-Schoenberg operators is considered. Their shape preserving properties are discussed. A simple sufficient condition for convergence of infinite sequences of Chebyshev-Schoenberg operators is established. The relations between Chebyshev-Bernstein operators and Chebyshev-Schoenberg operators are considered. The obtained results are compared with those obtained by \textit{M. J. Marsden} [J. Approximation Theory 3, 7--49 (1970; Zbl 0192.42103)].
scientific article; zbMATH DE number 5951122
scientific article