On Walsh's equiconvergence theorem and the application of summability methods (Q1322610)

Let \(f\) be regular in \(D_ R : = \{z : | z | < R\}\) for some \(R>1\) but not in \(\overline D_ R\), let \(L_ n\) be the Lagrange interpolating polynomial to \(f\) in the \((n+1)\)-th roots of unity, and let \(S_ n\) be the \(n\)-th partial sum of the Taylor expansion of \(f\) at 0. Walsh's equiconvergence theorem then states that \(L_ n (z) - S_ n(z) \to 0\) for every \(z\) with \(| z | < R^ 2\), uniformly on compact subsets. The author asks: Can we have equisummability in some larger disk \(\{z : | z | < \rho\}\) with \(\rho > R^ 2\)? More precisely: Let \(\lambda_ n (z) : = \sum^ \infty_{k = 0} a_{nk} L_ k(z)\) and \(\sigma_ n (z) : = \sum^ \infty_{k = 0} b_{nk} S_ k(z)\) with some infinite matrices \(A = (a_{nk})\) and \(B = (b_{nk})\). Can it happen that \(\lambda_ n (z) - \sigma_ n (z) \to 0\) for all \(z\) with \(| z | < \rho\), for some \(\rho > R^ 2\) and for all \(f\) under consideration? The answer is no under mild assumptions on \(A\) and \(B\) and certainly no if \(A\) and \(B\) are regular matrices.