A new basis for osculatory interpolation problems and applications

In this paper we present a polynomial basis based on two-point osculatory interpolation. By exploring some interesting properties of this basis, we derive the smoothness conditions. These conditions can be used for the construction of smooth splines with a low polynomial degree in terms of data points. As an application we give an efficient method for constructing composite splines with shape parameters.

[1]  G. Farin.  Curves and Surfaces for Cagd: A Practical Guide , 2001 .

[2]  R. E. Carlson,et al.  Monotone Piecewise Cubic Interpolation , 1980 .

[3]  Alexandru Mihai Bica,et al.  Fitting data using optimal Hermite type cubic interpolating splines , 2012, Appl. Math. Lett..

[4]  Larry L. Schumaker,et al.  Spline functions - computational methods , 2015 .

[5]  R. Peter Dube,et al.  Univariate Blending Functions and Alternatives , 1977 .

[6]  Carla Manni.  On Shape Preserving C2 Hermite Interpolation , 2001 .

[7]  Manabu Sakai,et al.  Osculatory interpolation , 2001, Comput. Aided Geom. Des..

[8]  R. E. Grundy.  Hermite interpolation visits ordinary two-point boundary value problems , 2007, The ANZIAM Journal.

[9]  M. Newman,et al.  Interpolation and approximation , 1965 .

[10]  Ahmed Tijini,et al.  A simple method for smoothing functions and compressing Hermite data , 2005, Adv. Comput. Math..

[11]  E. Passow,et al.  MONOTONE AND CONVEX SPLINE INTERPOLATION , 1977 .

[12]  A. K. Cline.  Scalar- and planar-valued curve fitting using splines under tension , 1974, Commun. ACM.

[13]  G. Phillips.  Explicit forms for certain Hermite approximations , 1973 .

[14]  Hans-Peter Seidel.  On Hermite interpolation with B-splines , 1991, Comput. Aided Geom. Des..

[15]  Larry I. Schumaker.  ON SHAPE PRESERVING QUADRATIC SPLINE INTERPOLATION , 1983 .

[16]  A. L. Perrie.  Uniform Rational Approximation with Osculatory Interpolation , 1970, J. Comput. Syst. Sci..