More quantitive results on Walsh equiconvergence: I. Lagrange case (Q580601)

Let \(A_{\rho}\) \((\rho >1)\) be the class of complex-valued functions, analytic in the disc \(| z| <\rho\) and having singularities on the boundary \(| z| =\rho\). For \(f\in A_{\rho}\) let \(L_{n-1}(f;z)\) be the Lagrange interpolation polynomial on the nth roots of unity and let \(Q_{n-1,\ell}(f;z)=\sum^{\ell -1}_{j=0}\sum^{n- 1}_{k=0}a_{k+jn}\cdot z^ k,\) where \((a_ k)\) are the Taylor coefficients of the function f. (If \(\ell =1\) then \(Q_{n-1,\ell}\) is the Taylor polynomial of degree n-1.) The author studies the order of convergence (or divergence) of the quantity \(\overline{\lim}_ n| \Delta_{\ell,n-1}(f;z)|^{1/n}\), where \(\Delta_{\ell,n- 1}(f;z)=L_{n-1}(f;z)-Q_{n-1,\ell}(f;z).\) The results obtained are also related to some problems studied by V. Totik, Quantitative results in the theory of overconvergence of complex interpolating polynomials (to appear in J. Approximation Theory).