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SAT 6 (2011), 2
Surveys in Approximation Theory, 6 (2011), 2474.
Uniform and Pointwise Shape Preserving Approximation by
Algebraic Polynomials
K. A. Kopotun, D. Leviatan, A. Prymak, and I. A. Shevchuk
Abstract.
We survey developments, over the last thirty years, in the theory of Shape
Preserving Approximation (SPA) by algebraic polynomials on a finite
interval. In this article, "shape" refers to (finitely many changes of)
monotonicity, convexity, or qmonotonicity of a function (for
definition,
see Section 4). It is rather well known that it is possible to approximate a
function by algebraic polynomials that preserve its shape (i.e., the
Weierstrass approximation theorem is valid for SPA). At the same time, the
degree of SPA is much worse than the degree of
best unconstrained approximation in some cases, and it is "about the same"
in others.
Numerous results quantifying this difference in degrees of SPA and
unconstrained approximation have been obtained in recent years, and the main
purpose of this article is to provide a "bird'seye
view" on this area, and discuss various approaches used.
In particular, we present results on the validity and invalidity of uniform
and pointwise estimates in terms of various moduli of smoothness. We compare
various constrained and unconstrained approximation spaces as well as orders
of unconstrained and shape preserving approximation of particular functions,
etc. There are quite a few interesting phenomena and several open questions.
Eprint: arXiv:1109.0968
Published: 29 August 2011.
K. A. Kopotun
Department of Mathematics
University of Manitoba
Winnipeg, Manitoba R3T 2N2
Canada
Email: kopotunk@cc.umanitoba.ca
D. Leviatan
Raymond and Beverly Sackler School of Mathematics
Tel Aviv University
Tel Aviv 69978
Israel
Email: leviatan@post.tau.ac.il
A. Prymak
Department of Mathematics
University of Manitoba
Winnipeg, Manitoba R3T 2N2
Canada
Email: prymak@cc.umanitoba.ca
I. A. Shevchuk
National Taras Shevchenko University of Kyiv
Kyiv, Ukraine
Email: shevchukh@ukr.net