Approximation of Zolotarev type (Q584452)

The authors use a simple example of an explicitly computable best complex polynomial approximation on the disk, which is actually an example where the Carathéodory-Fejér approximant is the best uniform approximation, for the construction of upper and lower bounds for the norm of generalized Zolotarev polynomials on [-1,1]. Given \(a\in {\mathbb{R}}\), \(k\in {\mathbb{N}}\), these are defined as polynomials of the form \(aT_{m+1+k}+T_{m+1+p}\) (where \(T_ n\) denotes the Chebyshev polynomial of degree n) with p a polynomial of degree m chosen so that the uniform norm on [-1,1] is minimum.