Based on Lebesgue's constant minimum skew-preserving asymptote of rational interpolation of the centre of gravity

Abstract

This study focuses on an in-depth investigation of centre of gravity rational interpolation as an important rational interpolation method. An important feature is that different interpolation weights can be used to derive different centre of gravity rational interpolation functions. By carefully choosing the interpolation weights, we can ensure that the centre of gravity rational interpolation does not produce poles or unreachable points, thus improving the numerical stability of the interpolation. However, when we try to approximate the interpolated function with oblique asymptotic properties, the traditional rational interpolation method may not be able to maintain the original oblique asymptotic properties. To cope with this problem, this paper proposes a novel centre of gravity rational interpolation method aimed at maintaining the oblique asymptote. First, we systematically investigate the conditions that the centre of gravity rational interpolation method needs to satisfy in order to ensure that the slant asymptote property is maintained. Then, we develop a complex optimisation model with the minimum value of Lebesgue's constant as the objective function, along with multiple constraints, including the rational interpolation of the centre of gravity not generating poles and unreachable points, the maintenance of the slanting asymptote property by the rational interpolation function of the centre of gravity, as well as the normalisation of the centre of gravity weights. By solving this model, we obtain the optimal values of the interpolation weights, which leads to the centre of gravity rational interpolation function possessing the slanting asymptote property. Through detailed numerical examples, we validate the effectiveness of this algorithm, demonstrating its excellent performance in approximating the interpolated function with oblique asymptotic properties. This research work not only provides new insights into theoretical mathematics, but also offers a more accurate and controllable interpolation method for engineering, science and data analysis, further promoting research and applications in related fields.