On a Fast Convergence of the Rational-Trigonometric-Polynomial Interpolation

We consider the convergence acceleration of the Krylov-Lanczos interpolation by rational correction functions and investigate convergence of the resultant parametric rational-trigonometric-polynomial interpolation. Exact constants of asymptotic errors are obtained in the regions away from discontinuities, and fast convergence of the rational-trigonometric-polynomial interpolation compared to the Krylov-Lanczos interpolation is observed. Results of numerical experiments confirm theoretical estimates and show how the parameters of the interpolations can be determined in practice. 1. Introduction In this paper, we continue investigations started in [1] where we considered the convergence acceleration of the classical trigonometric interpolation via polynomial corrections representing discontinuities of the function and some of its first derivatives (jumps). The resultant interpolation was called as the Krylov-Lanczos (KL-) interpolation. That approach was suggested in 1906 by Krylov [2] and later in 1964 by Lanczos [3, 4] (see also [1, 5–10] with references therein). Here, we consider the convergence acceleration of the KL-interpolation by the application of rational (by ) correction functions along the ideas of the rational approximations (see [11–13] with references therein). The approach discussed here leads to the parametric (depending on parameters ) rational-trigonometric-polynomial (rtp-) interpolation . The idea of the convergence acceleration via sequential application of polynomial and rational corrections was described in [14–17]. The KL-interpolation is a special case of the rtp-interpolation corresponding to the choice of parameters , . Besides, rational corrections can be applied immediately to the classical interpolation without polynomial corrections (see interpolation ). In this paper, we reveal the convergence properties of the rtp-interpolation, show its fast convergence compared to the KL-interpolation in the regions away from the singularities ( ), and discuss the problem of parameters determination in rational corrections. 2. Rational Interpolations In this section, we introduce a rational interpolation as a method of the convergence acceleration of the classical trigonometric interpolation. Here, we recap some details from [15, 16]. By , we denote the error of the classical trigonometric interpolation and write where is the th Fourier coefficient of Rational corrections considered in this paper are based on a series of formulae of summation by parts applied to the error terms in (4). Such transformations lead to new interpolations