On real and complex-valued bivariate Chebyshev polynomials (Q1262485)

The authors consider the following two problems: Problem A(B). Given arbitrary integers m,n with \(m+n\geq 1\), find all elements \(p^*\) of the space \(P_{m+n-1}\) of real (complex) bivariate polynomials of total degree \(m+n-1\) which best approximate the function \(f(x,y)=x^ ny^ m(f(w,z)=w^ nz^ m)\) on the unit square \(U=[-1,1]^ 2\) (on the biellipse \(B_{\phi}=E^ 2_{\phi}\), \(E_{\phi}=\{z\in {\mathbb{C}}| z=1/2(t+t^{-1})\), \(| t| =\phi \}\), \(1<\phi <\infty)\) in the uniform norm. The first contribution of this paper is to provide uniqueness conditions for Problem A, by exploiting the well-known Noether's theorem of algebraic plane curves theory and to discribe the set of best approximations in case of nonuniqueness. It is also proved in a simple manner that the product of monic Chebyshev polynomials of the first kind also yields an optimal error for Problem B and that the same uniqueness conditions prevail in the complex case.