On real and complex-valued bivariate Chebyshev polynomials

Abstract

In the real uniform approximation of the function x m y n by the space of bivariate polynomials of total degree m + n − 1 on the unit square, the product of monic univariate Chebyshev polynomials yields an optimal error. We exploit the fundamental Noether's theorem of algebraic curves theory to give necessary and sufficient conditions for unicity and to describe the set of optimal errors in case of nonuniqueness. Then, we extend these results to the complex approximation on biellipses. It turns out that the product of Chebyshev polynomials also provides an optimal error and that the same kind of uniqueness conditions prevail in the complex case. Yet, when nonuniqueness occurs, the characterization of the set of optimal errors presents peculiarities, compared to the real problem.