Treffer: weighted least square convergence of lagrange interpolation on the unit circle: Weighted least square convergence of Lagrange interpolation on the unit circle

Let \(Z_{2n}^{\alpha,\beta}\) be the set obtained by projecting vertically the zeros of the Jacobi polynomial \(P_n^{\alpha,\beta}\) onto the unit circle. Let \(L_n(f,z)\) denote the Lagrange interpolation polynomial for a given function \(f\) on the set \(Z_{2n}^{\alpha,\beta}\). When this set is augmented by one or both of the points \(\pm 1\) one gets four different cases. The author proves that for functions \(f\) analytic in the disc and continuous in \(|z|\leq 1\) the \(L^2\)-norm of \(|f(z)-L_n(f,z)|\) on the unit circle with the appropriate weight \(|1-z|^{2\alpha\pm 1}|1+z|^{2\beta\pm 1}\) converges to 0 as \(n \to \infty\). This is a generalization of a result of Walsh and Sharma on \(L^2\) approximation and interpolation in the roots of unity, which is included as the special case \(\alpha=\beta=\pm{1\over 2}\).