The Research Of The Maximum On The Lebesgue Function Of Barycentric Rational Interpolation

The advantages of barycentric interpolation formulations in computation are small number of floating point operations and good numerical stability.Adding a new data pair,the barycentric interpolation formula don't require renew computation all basis functions.As for fitting a large number of points,rational interpolation sometimes gives better approximations than polynomial interpolation,but it is difficult to control the occurrence of poles.The family of barycentric rational interpolants introduced by Floater and Hormann not only avoids the Runge's phenomenon for polynomial interpolation,but also covers the shortage of controlling the occurrence of poles for general rational interpolants,thus widely used in approximation theory and related religions.Firstly,the first two sections gives a general ideas about the research background,status quo and significance of BRI,and we briefly review the two main interpolation and each advantages and weaknesses,that is,Lagrange interpolation and rational interpolation.Secondly,we focus on stress Berrut's rational interpolant and Floater-Hormann barycentric rational interpolation,by introducing the Lebesgue and Lebesgue constant we compare each advantages and disadvantages.So far,the Lebesgue constant has been studied intensively in the case of Berrut's rational interpolant at equidistant nodes.In this paper,we extented many good properties of Berrut's rational interpolant and proved that the Lebesgue function possess symmetry and its sequences are strictly increasing over the interpolation interval.Berrut's rational interpolant is merely a special case of Floater-Hormann barycentric rational interpolation.In the interpolation function of Floater and Hormann,d decides on the weights and the properties of interpolants.As demonstrate in figures,different d declare different intervals where the function obtain its maximum.When d(28)2,we proved that the Lebesgue function obtains its maximum at two endpoints of the kont intervals.At the end of the article the function of BRI is improved,we also provide an extension of the Floater–Hormann family of barycentric rational interpolants.and we briefly study the interpolation problem in the case of equidistant nodes to quasi-equidistant ones.