Shape preserving representations and optimality of the Bernstein basis (Q1895889)

The paper deals with shape preserving properties of totally positive (TP) bases of polynomials of degree less than or equal to \(n\). A sequence of functions \(U:= (u_0, \dots, u_n)\) is TP on an interval \(I\) if for any \(t_0< t_1< \dots< t_m\) in \(I\) the collocation matrix \((u_j (t_i) )_{i=0, \dots, m, j=0, \dots, n}\) is TP, i.e., if all its minors are nonnegative. Given a sequence of positive functions \(U:= (u_0, \dots, u_n)\) on \([a, b]\) with \(\sum_{i=0}^n u_i (t)=1\) and a sequence \((C_0, \dots, C_n)\) of points in \(\mathbb{R}^k\), let \[ \gamma(t)= \sum_{i=0}^n C_i u_i (t) \qquad (t\in [a, b]). \] If \(U\) is TP, then the curve \(\gamma\) preserves many shape preserving properties of the control polygon generated by \((C_0, \dots, C_n)\). In the paper the following conjecture of \textit{T. N. Goodman} and \textit{H. B. Said} [Comput. Aided Geom. Des. 8, No. 2, 115-121 (1991; Zbl 0729.65006)]\ is proved: The Bernstein basis has optimal shape preserving properties among all normalized TB bases of polynomials of degree less than or equal to \(n\) over a compact interval. Further, a simple test for recognizing normalized totally positive bases with good shape preserving properties is proposed. Moreover, a corresponding corner cutting algorithm to generate the Bézier polygon is derived.