On an open problem of P. Turán concerning Birkhoff interpolation based on the roots of unity
Let f(z) be analytic in \((z)<1\) and continuous in (z)\(\leq 1\) and let \(z_{kn}\) \((k=1,...,n)\) denote the nth roots of unity. Let \(R_{n,m}(f;z):=\sum^{n}_{k=1}f(z_{kn})A_{k,n,m}(z)+\beta_{k,n, m}B_{k,n,m}(z)\) where \(A_{k,n,m}(z)\), \(B_{k,n,m}(z)\) are the fundamental polynomials of (0,m) interpolation on \(\{z_{kn}\}^ n_ 1\) and \(\beta_{k,n,m}=0(n^ m/\ell n n)\), \(k=1,...,n\). The main result of the authors is that \[ \lim_{n\to \infty}\frac{1}{2\pi}\int_{(z)=1}| f(z)-R_{n,m}(f;z)|^ 2| dz| =0 \] provided \(\beta_{kn}=0(n^ m)\), \(n=1,...,n\). This solves a problem proposed by \textit{P. Turán} in 1974 [J. Approximation Theory 29, 23-89 (1980; Zbl 0454.41001)] for the special case \(m=2\).
scientific article; zbMATH DE number 3964599
scientific article