Institution: Stanford University, Department of Computer Science

Title: Chebyshev polynomials are not always optimal

Author: Fischer, Bernd

Author: Freund, Roland

Date: June 1989

Abstract: We are concerned with the problem of finding among all polynomials of degree at most n and normalized to be 1 at c the one with minimal uniform norm on Epsilon. Here, Epsilon is a given ellipse with both foci on the real axis and c is a given real point not contained in Epsilon. Problems of this type arise in certain iterative matrix computations, and, in this context, it is generally believed and widely referenced that suitably normalized Chebyshev polynomials are optimal for such constrained approximation problems. In this note, we show that this is not true in general. Moreover, we derive sufficient conditions which guarantee that Chebyshev polynomials are optimal. Also, some numerical examples are presented.

http://i.stanford.edu/pub/cstr/reports/cs/tr/89/1264/CS-TR-89-1264.pdf