Tchebycheff polynomials on a disk.
There is a well-known characterization of the Chebychev polynomials of the first kind \(T_n\) \[ \| x^n-q_{n-1}\| _{[-1,1]}\geq 2^{1-n}\| T_n\| _{[-1,1]}, \] where \({\| \cdot\| _{[-1,1]}}\) is the uniform norm on the segment \({[-1,1]}\), \(q_n\) is an arbitrary polynomial of degree\({\;\leq n-1}\). Let \(\Pi^2_n\) be the space of polynomials in two variables with total degree\({\;\leq n}\). Let \(D\) be the unit disk on the plane. The authors define the Chebychev polynomials \(T_{n,m}\) on \(D\) and show that \[ \| x^m y^n-q_{m,n}(x,y)\| _{D}\geq 2^{1-n-m}\| T_{m,n}\| _{D}, \] where \({q_{m,n}\in\Pi^2_{n+m-1}}\). The Chebychev polynomials \(T_{n,m}\) are expressed explicitly by the classical Chebychev polynomials of first and second kind.
scientific article; zbMATH DE number 2032631