On the weighted Grünwald–Rogosinski process

Abstract. It is known that for every set of interpolation nodes, there exists a continuous function for which the sequence of Lagrange interpolation polynomials is not uniformly convergent. In the case of the Chebyshev abscissas, G. Grünwald constructed a process that is uniformly convergent for all continuous functions on the whole interval \([-1,1]\). However, M.S. Webster showed that for the roots of the Chebyshev polynomials of the second kind, the analogous construction is uniformly convergent only in closed subintervals of \((-1,1)\). Our aim is to improve this result by using weighted Lagrange interpolation. We shall prove that the weighted Grünwald–Rogosinski process is uniformly convergent on the whole interval \([-1,1]\) in suitable weighted function spaces. Order of convergence will also be investigated.