Piecewise Chebyshev approximation. Part 1: Brief Description

As I said I will try to solve a problem of optimal piecewise Chebyshev approximation. What the problem is?
We have -- any computable function, defined at . It may be non-continuous. Also we have some -- a number of approximation partitions. So, we also have a set of intervals: , containing boundaries for each part of partition. On each of these intervals must exist a constant function , and this function must be a Chebyshev approximation of on .
Let's take a look at a single interval . For uniform approximation Chebyshev alternance condition must be satisfied. So, let's define function . Satisfying Chebyshev alternance we must say that has two extremums and values of these extremums are equal, while their signs differ. Absolute value of these extremums we'll denote as . Looking at piecewise approximation problem at a whole we can reformulate it as a problem of getting equal at all of intervals. About solving this problem read following posts.