代数多项式的均匀和逐点保形逼近
Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials
AI总结：
本文综述了近三十年来有限区间上代数多项式保形逼近理论的发展，讨论了均匀和逐点估计的有效性与无效性，并比较了不同约束和无约束逼近空间及特定函数的逼近阶。
摘要：
我们综述了近三十年来有限区间上代数多项式保形逼近（SPA）理论的发展。在本文中，“形状”指函数的单调性、凸性或q-单调性（有限次变化）（定义见第4节）。众所周知，可以用代数多项式逼近函数并保持其形状（即Weierstrass逼近定理对SPA成立）。同时，在某些情况下SPA的逼近度远差于最佳无约束逼近度，而在其他情况下则“大致相同”。近年来，量化SPA与无约束逼近度差异的大量结果已被获得，本文的主要目的是提供该领域的“鸟瞰图”，并讨论所采用的各种方法。特别地，我们给出了关于各种光滑模的均匀和逐点估计的有效性与无效性的结果。我们比较了各种约束和无约束逼近空间，以及特定函数的无约束和保形逼近阶等。存在相当多的有趣现象和几个未解决问题。
英文摘要：
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 q-monotonicity 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's-eye 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
