Shape-Preserving Approximation by Real and Complex Polynomials

Abstract

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On saturation effect for linear shape-preserving approximation in Sobolev spaces

“…The part of the approximation theory, which is devoted to these kind of problems, is known as the theory of shape preserving approximation (see e.g. [3,9] for a survey).…”

BASIC PROPERTIES OF LINEAR RELATIVE n-WIDTHS

“…This theory studies approximation properties of the different methods for approximation of functions preserving its shape properties (monotonicity, convexity). The most significant results were gathered in [1], [2]. Note that if a function f has some shape properties, it usually means that element f belongs to a cone (a convex set, closed under nonnegative scalar multiplication).…”

Approximation of fuzzy numbers by nonlinear Bernstein operators of max-product kind

“…The main aim of this note is to use the so-called Bernstein operator of max-product kind, firstly introduced (and formally studied) in the book [14], p. 325-326 and completely studied in the papers [9], [8], [13], for approximating fuzzy numbers with continuous membership functions.…”

“…For a positive continuous function f : [0, 1] → R, the max-product Bernstein operator was defined in [14], by Notice that the max-product Bernstein operator is obtained from the linear Bernstein polynomial written in the form…”