交替定理与切比雪夫样条逼近
Alternance Theorems and Chebyshev Splines Approximation
AI总结：
本文通过新方法证明了切比雪夫多项式逼近的最优性条件，并提出基于此的构造性算法，同时发展了自由结样条逼近的局部最优条件。
摘要：
本文旨在通过新方法证明切比雪夫多项式逼近和分段多项式逼近问题中的交替性质。在本文的第一部分中，我们提出了一种原始方法，以获得已知的必要和充分最优性条件的新证明。这种方法有两个主要优势。首先，证明比现有证明更简单且更容易理解。其次，这些证明是构造性的，因此导致了基于交替的算法，可以被视为Remez型逼近算法。在本文的第二部分中，我们为自由结多项式样条逼近发展了新的局部最优条件。自由结逼近的证明依赖于第一部分中开发的技术。
英文摘要：
One of the purposes in this paper is to provide a better understanding of the alternance property which occurs in Chebyshev polynomial approximation and piecewise polynomial approximation problems. In the first part of this paper, we propose an original approach to obtain new proofs of the well known necessary and sufficient optimality conditions. There are two main advantages of this approach. First of all, the proofs are much simpler and easier to understand than the existing proofs. Second, these proofs are constructive and therefore they lead to alternative-based algorithms that can be considered as Remez-type approximation algorithms. In the second part of this paper, we develop new local optimality conditions for free knot polynomial spline approximation. The proofs for free knot approximation are relying on the techniques developed in the first part of this paper.
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。
