Variation diminishing properties of Bernstein polynomials on a tetrahedron
The authors extend to \(\mathbb{R}^ d\) various definitions of the notions of total variation of a function \(f\) defined on a simplex. The original definitions were given by \textit{T. N. T. Goodman} for functions defined on a triangle [see J. Approximation Theory 50, 111-126 (1987; Zbl 0618.41007) and Constructive Approximation 3, 297-305 (1987; Zbl 0634.41020)]. Estimates are given on the total variation of the \(n\)-th Bernstein polynomial \(B_ n(f)\), of the function by means of the total variation of the \(n\)th Bézier net \(f_ n\) associated with the function. The constants depend on \(n\) and \(d\) and are given explicitly. These estimates extend those obtained by Goodman in the above mentioned papers.
scientific article; zbMATH DE number 701328