The Approximation Property Of Barycentric Rational Interpolant At Well-spaced Nodes

Barycentric rational interpolation is a new research hot spot in the field of approxima-tion theory and geometric modeling due to its minor calculation and no pole.This paper focuses on the approximation properties of two kinds of Berrut's rational interpolation at several kinds of well-spaced nodes,the main research results include the f'ollowing three aspects:By introducing q-integers as scaling factors,this paper constructs a new kind of affine interpolation nodes——q-equidistant nodes,furthermore,it is proved that q-equidistant nodes are well-spaced nodes when scaling factors q?(1-1/n-1/n2,1+1/n+1/n2).To some extent,this paper corrects the conclusion,which Cirillo and Hormann point out in their 2017 paper that affine nodes not be well-spaced nodes.Lebesgue constant is an important tool for measuring the numerical stability of inter-polation operators.Corresponding to two different value ranges of the scaling factor q,this article derives the upper bound of the Lebesgue constant for Berrut's rational interpolation of the first kind at q-equidistant nodes,it indicates the upper bound of the Lebesgue con-stant for Berrut's rational interpolation of the first kind at q-equidistant nodes increases logarithmically with regard to the interpolation nodes n and it is numerically stable when q ?(1-1/n-1/n2,1)and q?(1,1+1/n+1/n2).And compared with symmetric affine nodes,it is verified that Berrut's rational interpolation of the first kind at q-equidistant nodes has a smaller upper bound of the Lebesgue constant,which means we achieve better numerical stability.Berrut's rational interpolation operator of the second kind is barycentric for any n,which can guarantee that the interpolation is linear regeneration,and Berrut's rational in-terpolation operator of the second kind is a modification of Berrut's rational interpolation operator of the first kind that only is barycentric for odd n.This paper studies the approxi-mation properties of Berrut's rational interpolation of the second kind at equidistant nodes,quasi-equidistant nodes and well-spaced nodes,we prove the upper bounds of the Lebesgue constant of the three kinds of interpolation nodes increase logarithmically with regard to of n,which means we achieve better numerical stability.Compare to what we know,the upper bound of the Lebesgue constant of Berrut's rational interpolation of the second kind at quasi-equidistant nodes is strengthened;Compared to Berrut's rational interpolation of the first kind,Berrut's rational interpolation of the second kind has a smaller upper bound of Lebesgue constant at well-spaced nodes,which has the better approximation properties.