Rate of convergence for lacunary interpolation processes based on the roots of unity (Q2639987)

Let \(z_ k\) (k-1,...,n) be the nth roots of unity and let f be analytic on the unit disk D and continuous on \(\bar D.\) The authors investigate the behaviour of \[ ([2\pi^{-1}\int_{| z| =1}| f(z)-R_ n(z)|^ 2| dz|]^{1/2}, \] where \(R_ n\) is a polynomial of degree \(\leq 3n-1\) satisfying \(R_ n(z_ k)=f(z_ k)\) \(k=1,...,n)\) and obeying certain estimates for the derivatives \(R'_ n(z_ k)\), \(R'''_ n(z_ k)\) \((k=1,...,n)\). The results are closely related to \textit{J. Szabados} and \textit{A. K. Varma} [J. Approximation Theory 47, 255- 264 (1986; Zbl 0598.41003)].