On a Pointwise Convergence of Quasi-Periodic-Rational Trigonometric Interpolation

We introduce a procedure for convergence acceleration of the quasi-periodic trigonometric interpolation by application of rational corrections which leads to quasi-periodic-rational trigonometric interpolation. Rational corrections contain unknown parameters whose determination is important for realization of interpolation. We investigate the pointwise convergence of the resultant interpolation for special choice of the unknown parameters and derive the exact constants of the main terms of asymptotic errors. 1. Introduction The quasi-periodic (QP) interpolation , ( is integer) and , interpolates function on equidistant grid and is exact for a quasi-periodic function with period which tends to as . The idea of the QP interpolation is introduced in [1, 2] where it is investigated based on the results of numerical experiments. Explicit representation of the interpolation is derived in [3–5]. There, the convergence of the interpolation is considered in the framework of the -norm and at the endpoints in terms of the limit function. Pointwise convergence in the interval is explored in [6]. The main results there, which we need for further comparison, are the following theorems. Let We denote by the error of the QP interpolation as follows: Theorem 1 (see [6]). Let for some , , and Then, the following estimate holds for as where Theorem 2 (see [6]). Let for some and Then, the following estimate holds for as : In the current paper, we consider convergence acceleration of the QP interpolation by rational corrections in terms of which leads to quasi-periodic-rational (QPR) interpolation. We investigate the pointwise convergence of the QPR interpolation in the interval and derive the exact constants of the main terms of asymptotic errors. Comparison with Theorems 1 and 2 shows the accelerated convergence for smooth functions. Some results of this research are reported also in [7]. More specifically, the QP interpolation can be realized by the following formula: where Here, are the elements of the inverse of the Vandermonde matrix as and have the following explicit form [8]: where are the coefficients of the following polynomial: Taking into account that , from (13), we get 2. Quasi-Periodic-Rational Interpolation In this section, we consider convergence acceleration of the QP interpolation by rational trigonometric corrections which leads to the QPR interpolation. Consider a vector . By , we denote generalized finite differences defined by the following recurrent relations: for some sequence . When , we put It is easy to verify that In general, we can prove by the