Interpolation And Approximation Of Rational Spline

This paper studies the interpolation theory of univariate rational spline, which constructs chiefly a kind of rational spline with prescribed poles and Pade-type spline, then presents its way of construction, nature and representative. Main contents of the paper are listed as following three parts:Firstly, we review the history of rational spline, and present definition, representation theorems and some generalizations about classifiable rational interpolation spline.Secondly, by analyzing the theories of classifiable rational interpolation spline, and using relevant knowledge of Padé approximation, we estimate the poles' range of a meromorphic function. Furthermore, using the information of prescribed poles, we construct a kind of rational spline with prescribed poles, present its algorithm and the proof of its uniqueness, numerical example and graph explain its superiority.Lastly, based on the relative theories of Padé spline and Padé-type approximation, we construct Padé-type spline using both function values and up to k order derivative values of the function being interpolated as the interpolation data, prove its uniqueness, present its way of construction, its numerical example, and its graph. The Pade-type spline we construct can not only select denominators by the specialty of interpolated function to have good approximation effect, but also avoid solving higher non-linear equations. The graph shows that the Padé-type spline gives a better approximation to the function being interpolated than the Padé spline does.