Symmetric multivariate Chebyshev polynomials

We introduce a general method for the symmetrization of univariate polynomials and use it to construct symmetric polynomials in r + 1 variables that generalize the classical Chebyshev polynomials of the first kind. We show that, on the set [−1, 1]r+1, such polynomials provide the best uniform approximations of the complete symmetric polynomials hλ(x0, x1, …, xr) by means of symmetric polynomials of total degree less than n, where λ is a partition of n.