On a pointwise convergence of trigonometric interpolations with shifted nodes

Abstract

We consider trigonometric interpolations with shifted equidistant nodes and investigate their accuracies depending on the shift parameter. Two different types of interpolations are in the focus of our attention: the Krylov-Lanczos and the rational-trigonometric-polynomial interpolations. The Krylov-Lanczos interpolation performs convergence acceleration of the classical trigonometric interpolation by polynomial corrections. Additional acceleration is achieved by application of rational corrections which contain some extra parameters. In both cases, we derive the exact constants of the asymptotic errors and, based on these estimates, we find the optimal shifts that provide with the best accuracy. Optimizations are performed for the pointwise convergence in the regions away from the endpoints. Asymptotic estimates allow optimal selection of the extra parameters in the rational corrections which provides with additional accuracy. Results of numerical experiments clarify theoretical investigations.