Converse results in the theory of equiconvergence of interpolating rational functions (Q910514)

Let \(\rho >1\) and a function f analytic in \(| z| <\rho\) be given. For a fixed \(\sigma >1\) and a fixed integer \(m\geq -1\) define rational functions \(R_{n+m,n}(z,f):=B_{n+m,n}(z,f)/(z^ n-\sigma^ n)\), where \(B_{n+m,n}(z,f)\) is the Lagrange interpolant (of degree \(\leq n+m)\) to \((z^ n-\sigma^ n)f(z)\) in the \((n+m+1)\)-th roots of unity. With certain rational functions \(r_{n+m,n}(z,f,\nu)=P_{n+m,n}(z,f,\nu)/(z^ n-\sigma^ n)\) the deviation \[ \Delta^{\sigma}_{\ell,n,m}(z;f)=R_{n+m,n}(z,f)- \sum^{\ell -1}_{\nu =0}r_{n+m,n}(z,f,\nu) \] is defined (for a fixed positive integer \(\ell)\). The (difficult) construction of the polynomials \(P_{n+m,n}(z,f)\) uses Hermite interpolation in the zeros of \([(z^{m+1}(z^ n-\sigma^{-n})]^{\nu +1}\) and a relation due to \textit{E. B. Saff} and \textit{A. Sharma} [Rational approximation and interpolation, Proc. Conf., Tampa/Fla. 1983, Lect. Notes Math. 1105(1984), 256-271 (1984; Zbl 0558.41007)], who have shown that if \(\sigma \geq \rho^{\ell +1}\) then \(\lim_{n\to \infty}\Delta^{\sigma}_{\ell,n,m}(z,f)=0\) for \(| z| <\rho^{\ell +1}.\) The present author gives a partial converse of this statement: If f is analytic in \(| z| <1\) and continuous in \(| z| \leq 1\), if for some \(\rho >1\) and for some \(\sigma >\rho^{\ell +1}\) the sequence \(\{\Delta^{\sigma}_{\ell,n,n}(z,f)\}^{\infty}_{n=1}\) is uniformly bounded in every closed subset of \(| z| <\rho^{\ell +1}\), then f is analytic in \(| z| <\rho\). This is an analogue of a result due to \textit{J. Szabados} [Analysis 2, 267-280 (1982; Zbl 0532.30002)] who considered certain polynomials in the places of the rational functions \(R_{n+m,n}\) and \(r_{n+m,n}.\) \{Misleading misprints: In (1.1) and (3.1), r should read R.\}