On the constrained Chebyshev approximation problem on ellipses
The following constrained Chebyshev approximation problem on ellipses is considered: \[ D_ n(r,c):=\min_{p\in \prod_ n,p(c)=1}\max_{z\in {\mathcal E}_ r}| p(z)|, \] where \(\prod_ n\) denotes the set of all complex polynomials of degree at most n \[ {\mathcal E}_ r=\{z\in {\mathbb{C}};| z-1| +| z+1| \leq r+\frac{1}{r}\}r\geq 1, \] is an ellipse in the complex plane \({\mathbb{C}}\) with foci at \(\pm 1\), and it is always assumed that \(c\in {\mathbb{C}}\setminus {\mathcal E}_ r\). The solution of (1) is classical for the case \(r=1\), of the line segment \({\mathcal E}_ 1=[-1,1]\) real c: \(P_ n(z,r,c)=T_ n(z)/T_ n(c),\) where \(T_ n\) is the nth Chebychev polynomial. In this paper the solution of (1) on \({\mathcal E}_ r\) is characterized, founded and studied. This problem were also studied by \textit{G. Opfer} and \textit{G. Schober} [Linear algebra appl. 58, 343-361 (1984; Zbl 0565.65012)] for \(n=1\) with \({\mathcal E}\subset {\mathbb{C}}\) any compact set not containing c.
Chebyshev approximation problem on ellipses
scientific article; zbMATH DE number 4202989