Polynomial spaces revisited via weight functions

167-198International audienceExtended Chebyshev spaces are natural generalisations of polynomial spaces due to the same upper bounds on the number of zeroes. In a natural approach, many results of the polynomial framework have been generalised to the larger Chebyshevian framework, concerning Approximation Theory as well as Geometric Design. In the present work, we go the reverse way: considering polynomial spaces as examples of Extended Chebyshev spaces, we apply to them results specifically developed in the Chebyshevian framework. On a closed bounded interval, each Extended Chebyshev space can be defined by means of sequences of generalised derivatives which play the same rôle as the ordinary derivatives for polynomials. We recently achieved an exhaustive description of the infinitely many such sequences. Surprisingly, this issue is closely related to the question of building positive linear operators of the Bernstein type. As Extended Chebyshev spaces, one can thus search for all generalised derivatives which can be associated with polynomials spaces on closed bounded intervals. Though this may a priori seem somewhat nonsensical due to the simplicity of the ordinary derivatives, this actually leads to new interesting results on polynomial and rational Bernstein operators and related results of convergence

Evaluarea ordinului de aproximare prin operatori liniari si pozitivi

Auswertung der Approximationsgüte durch positive lineare Operatoren Abstract: The areas of research covered by this thesis can be roughly divided into three parts: aspects of quantitative approximation, studying of shape-preservation properties and over-iteration for some selected operators. Regarding the quantitative approximation we study traditional problems like: direct estimates, degree of simultaneous approximation or global smoothness preservation (Chapters 2, 3). On the other hand, we also present some non-classical issues like: estimates for the Peano remainder, a quantitative Voronovskaja theorem or estimates for differences of two positive linear operators (Chapter 5). Our object of study are different classes of operators: rational type operators, composite Beta type operators, but also other types that cannot be classified like: the BLaC-wavelet operator and the King operator

A family of univariate rational Bernstein operators

International audienceWe define and study a new family of univariate rational Bernstein operators. They are positive operators exact on linear polynomials. Moreover, like classical polynomial Bernstein operators, they enjoy the traditional shape preserving properties and they are total variation diminishing. Finally, for a specific class of denominators, some convergence results are proved, in particular a Voronovskaja theorem, and some error bounds are given