Bivariate polynomials of least deviation from zero (Q2715680)

The polynomial \(\omega_{m,n}\) of the form \( \omega_{m,n}(x,y)=x^my^n-p(x,y),\) with \(p\) of degree \(m+n-1,\) is said to be an error function of type \(\{m,n\}\) on a set \(\Omega\subseteq\mathbb R^2,\) with respect to a given \(L^p\) norm, \(p\in [1,\infty],\) if \(\|\omega_{m,n}\|_{\Omega,p}\) is minimal. The authors obtain sample of error functions \(\omega_{m,n},\) with respect to uniform or \(L^2\) norm on the unit disk \(D\subseteq\mathbb R^2\) (on the triangle \(S: x\geq 0,y\geq 0,x+y\leq 1,\) respectively), in terms of Chebychev polynomials of the first and the second kind. For example, NEWLINE\[NEWLINEQ_{m,n}(x,y)=\frac 1{2^{2(m+n)}}\{[U_m(2x-1)+U_{m-1}(2x-1)] [U_n(2y-1)+U_{n-1}(2y-1)]NEWLINE\]NEWLINE NEWLINE\[NEWLINE+[U_{m-1}(2x-1)+U_{m-2}(2x-1)]\cdot [U_{n-1}(2y-1)+U_{n-2}(2y-1)]\},NEWLINE\]NEWLINE \(U_n\) being Chebychev polynomials of the second kind on [-1,1], and an error function of type \(\{m,n\}\) on \(S\) with respect to uniform norm. A similar result is obtained on \(D\) with respect to the \(L^p\) norm in the set of products of linear bivariate polynomials.