Finite alternation theorems and a constructive approach to piecewise polynomial approximation in Chebyshev Norm

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.