Barycentric Rational Hermite Interpolation Based On The Lebesgue Constant Minimizing

From ancient times to the age of information and communication, both for astronomy or used in signal and image processing, interpolating is always extensive used in many areas of technology. Lagrange interpolating polynomial, Newton interpolating polynomial and Hermite interpolating polynomial is the most common of several traditional interpolation methods, but due to the high order polynomial may arise Runge phenomenon, limits the application of polynomial interpolation. Rational interpolation is used, when the interpolation nodes excessive. But the rational interpolation is hard to avoid the poles and unattainable points. Based on the rational interpolation, relaxation of restrictions on the degree of the numerator and denominator of rational interpolation, Barycentric rational Hermite interpolation is not only good numerical stability, small calculation, but also avoids the pole and unreachable point. There are many ways to find the weights, different weights lead different Barycentric rational Hermite interpolation. Based on Barycentric rational Hermite interpolation method, construct the Lebesgue constants of univariate Barycentric rational Hermite interpolation, Barycentric rational Hermite interpolation method based on the minimizing Lebesgue constant. Specifically, the minimizing Lebesgue constant is given for the objective function, the weights as decision variables, at the same time the interpolant should satisfy the interpolation conditions, on poles and unattainable points to establish the optimization model, get the optimal interpolation weights. On this basis, add some shape-preserving constraints, we make shape-preserving Barycentric rational Hermite interpolation. Finally, the bivariate Barycentric rational Hermite interpolation based on the Lebesgue constant minimizing, and proved the effectiveness of the new method by numerical examples.