On the order of approximation for the rational interpolation to |x|

Abstract

The order of approximation for Newman-type rational interpolation to |x| is studied in this paper. For general set of nodes, the extremum of approximation error and the order of the best uniform approximation are estimated. The result illustrates the general quality of approximation in a different way. For the special case where the interpolation nodes are\(x_i = \left( {\frac{i}{n}} \right)^r (i = 1,2, \cdots ,n;r > 0)\), it is proved that the exact order of approximation is\(O\left( {\frac{1}{n}} \right),O\left( {\frac{1}{{n\log n}}} \right) and O\left( {\frac{1}{{n^r }}} \right)\), respectively, corresponding to 0 1.