Convergence of certain polynomial interpolants to a function defined on the unit circle
Let f be analytic on the unit disk D and let \(f^{(r-1)}\) exist and be Riemann integrable on \(\partial D\). We denote by \(h_{rn-1}\) the polynomial of degree rn-1 which satisfies \[ h^{(j)}_{rn-1}(w_ k)=f^{(j)}(w_ k),\quad w_ k=e^{2\pi ik/n} \] for \(k=1,...,n\); \(j=0,...,r-1\). The authors show the compact convergence of \(h_{rn- 1}(z)\) to \[ (2\pi i)^{-1}\int_{| w| =1}f(w)/(w-z)dw \] in D. They obtain two corresponding results replacing \(h_{rn-1}\) by other polynomials which are characterized either by \(\ell_ 2\)-approximation or by the average of partial sums of \(h_{rn-1}\).
scientific article; zbMATH DE number 4105308