A note on certain next-to-interpolatory rational functions (Q1310431)

Let \(m\geq 0\) be a fixed integer, and let \(r_{n+ m-1} (z,f)\) be a rational function of the form \[ {p(z)\over z^ n-\sigma^ n},\quad p(x)\in \pi_{n+ m-1},\quad \sigma>1.\tag{1} \] It is known that if \(f\in A_ \rho\) (analytic in \(| z|<\rho\), but not in \(| z|\leq\rho\)), \(\rho>1\), then \(r_{n+ m-1}(z)\) minimizes \(\int_{| z|=1} | f(z)- r(z)|^ 2 dz\) over all \(r(z)\) of the form (1) if and only if \(r_{n+ m-1}(z)\) interpolates \(f(z)\) in the zeros of \((z^ n- \bar \sigma^ n) z^ m\). Here the author considers rationals \(R_{n+ m-1}(z,f)\) of the form (1) which minimize \(\max_{0\leq k\leq n+m} | f(\omega^ k)- R(\omega^ k)|\) over all rationals \(R(z)\) of the form (1). He shows for the difference \[ \lim_{n\to\infty} \bigl\{R_{n+ m-1} (z,f)- r_{n+m- 1}(z,f)\bigr\}=0\;\begin{cases}\text{if } | z|< \rho^ 2\quad(\sigma\geq\rho^ 2)\\\text{if }| z|=\sigma\quad(\sigma<\rho^ 2)\end{cases} \] the convergence being uniform and geometric on any compact subset of the regions above. Moreover, the result is shown to be sharp. When \(m=-1\), the author recovers theorem 2 in a paper of \textit{A. S. Cavaretta jun.}, \textit{An. Sharma} and \textit{R. S. Varga} [Interpolation in the roots of unity: An extension of a theorem of J. L. Walsh, Result. Math. 3, 155-191 (1980; Zbl 0447.30020)].