We know that the monic Chebyshev polynomial of degree n, T_n(x), has the least deviation from the origin among all polynomials of degree n. What about the polynomial of degree n that has the least deviation from the origin that also satisfies boundary conditions, continuity at both end points, for example?
The problem can be explained for a particular case as follows:
Suppose that we are interested in approximating a polynomial f of degree >n on [-1,1] by a polynomial p of degree n so that they meet at the endpoints.
Minimize Maximum |f(x)-p(x)| on [-1,1].