The Approximation Property Of Berrut's Rational Interpolation At Two Types Of Well-Spaced Nodes

A collection of recent papers reveals that barycentric rational interpolation,for example Berrut's rational interpolation,is a good choice for approximating smooth functions,especially when interpolating at well-spaced nodes,whose Lebesgue constant grows logarithmically with the number of nodes.Since the nodes generated by regular distribution functions are well-spaced,we focus on two kinds of regular distribution functions by using M(?)bius transformation on variables and functions,respectively.The regularity of these two kinds of new distribution functions and the approximation property of Berrut's rational interpolation at the new-generated well-spaced nodes are studied,and the main results are as follows:We make M(?)bius transformations on the variables of the function,and construct the -kind distribution function with convexity and degeneration,and prove they are regular.Moreover,we construct symmetric -kind regular distribution function based on symmetry and give the upper bound of the Lebesgue constant.Finally,numerical experiments show that the interpolation approximation property of -logarithmic nodes,symmetric -equidistant nodes and symmetric -logarithmic nodes are superior to classical equidistant nodes,and the approximation of symmetric -logarithmic distribution nodes is optimal.We also make the M(?)bius transformation on functions and construct another kind of distribution function,quasi -kind distribution function,that involves convexity and degeneration.We construct symmetric quasi -kind distribution function based on symmetry,we also prove the quasi -kind distribution function and the symmetric quasi -kind distribution function are regular.And we also give the upper bound of the Lebesgue constant.The numerical experiments show that the approximation property of the quasi-kind well-spaced nodes and the symmetric quasi -kind well-spaced nodes are superior to equidistant nodes and compare them with the symmetric -logarithmic nodes.When is in a certain range,the symmetric quasi -logarithmic nodes is best.