Converse results on equiconvergence of interpolating polynomials
Let \(U_ n:=\{e^{2\pi ik/n}\), \(k=0,...,n-1\}\), \(U=\cup^{\infty}_{n=1}U_ n\) be the roots of unity and \((a_ k)\) be a null sequence of complex numbers. Let f be defined on U and \[ \Delta_{\ell,n-1}(z):=L_{n-1}(z)-\sum^{n-1}_{k=0}\sum^{\ell - 1}_{j=0}a_{k+jn}z^ k,\quad \ell \in {\mathbb{N}}, \] where \(L_{n-1}\) is the Lagrange interpolant to f on \(U_ n\). The authors \(prove:\) (\(\Delta_{\ell,n-1})\) is uniformly bounded on compact subsets of \(| z| <\rho^{1+\ell}\) if and only if \(g(z):=\sum^{\infty}_{k=0}a_ kz^ k\) is analytic in \(| z| <\rho\) and there exists a function h analytic in \(| z| <\rho^{1+\ell}\) such that \(g(z)+h(z)=f(z)\) for \(z\in U\). This theorem contains a result of \textit{J. Szabados} [Analysis 2, 267-280 (1982; Zbl 0532.30002)].
scientific article; zbMATH DE number 4089865