Comparison of the Best Uniform Approximations of Analytic Functions in the Disk and on Its Boundary

A. A. Pekarskii

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Function spaces, approximation theory, and nonlinear analysis, Tome 255 (2006), pp. 227-232

Résumé

Denote by $C_A$ the set of functions that are analytic in the disk $|z|1$ and continuous on its closure $|z|\le 1$; let $\mathcal {R}_n$, $n=0,1,2,\dots$, be the set of rational functions of degree at most $n$. Denote by $R_n(f)$ ($R_n(f)_A$) the best uniform approximation of a function $f\in C_A$ on the circle $|z|=1$ (in the disk $|z|\le 1$) by the set $\mathcal {R}_n$. The following equality is proved for any $n\ge 1$: $\sup \{R_n(f)_A/R_n(f)\colon f\in C_A\setminus \mathcal {R}_n\}=2$. We also consider a similar problem of comparing the best approximations of functions in $C_A$ by polynomials and trigonometric polynomials.