On Rational Interpolation to |x| at the Adjusted Chebyshev Nodes

✍ Scribed by Lev Brutman

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 167 KB
• Volume: 95
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1998.3206

✦ Synopsis

Recently Brutman and Passow considered Newman-type rational interpolation to |x| induced by arbitrary sets of symmetric nodes in [&1, 1] and showed that under mild restrictions on the location of the interpolation nodes, the corresponding sequence of rational interpolants converges to |x|. They also studied the special case where the interpolation nodes are the roots of the Chebyshev polynomials and proved that for this case the exact order of approximation is O(1Ân log n), which, in view of Werner's result, is the same as for rational interpolation at equidistant nodes. In the present note we consider the set of interpolation nodes obtained by adjusting the Chebyshev roots to the interval [0, 1] and then extending this set to [&1, 1] in a symmetric way. We show that this procedure improves the quality of approximation, namely we prove that in this case the exact order of approximation is O(1Ân 2 ).