On polynomial function approximation with minimum mean squared relative error and a problem of Tchebychef

We consider the problem of constructing a polynomial approximation to a function f(x)f(x) over the interval [-1,1][-1,1] that minimizes the mean squared relative error (MMSRE) over the interval. We establish sufficient conditions for solving the problem. We then consider a classic problem from a paper of Tchebychef and compare his solution to MMSRE, demonstrating that in some cases the latter approach can yield a more appealing solution and one that it is applicable in a number of situations where the Tchebychef approach is not.