One of the purposes in this paper is to provide a better understanding of the alternance property which occurs in Chebyshev polynomial approximation and continuous piecewise polynomial approximation problems. In the first part of this paper, we prove that alternating sequences of any continuous function are finite in any given segment and then 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 intuitive and easy to understand. Second, these proofs are constructive and therefore they lead to new alternation-based 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. The piecewise polynomials are required to be continuous on the approximation segment.

中文翻译：

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Chebyshev范数中分段多项式逼近的有限交替定理和构造方法

本文的目的之一是提供对Chebyshev多项式逼近和连续分段多项式逼近问题中出现的交流特性的更好理解。在本文的第一部分中，我们证明了任何连续函数的交替序列在任何给定段中都是有限的，然后提出了一种原始方法来获得众所周知的必要条件和充分最优条件的新证明。这种方法有两个主要优点。首先，证明是直观且易于理解的。其次，这些证明是建设性的，因此导致了新的基于交替的算法。在本文的第二部分，我们为自由结多项式样条逼近开发了新的局部最优条件。自由结近似的证明依赖于本文第一部分中开发的技术。分段多项式要求在逼近段上是连续的。