  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.
