Abstract
We are concerned with the problem of finding the polynomial with minimal uniform norm on E among all polynomials of degree at most n and normalized to be 1 at c. Here, E is a given ellipse with both foci on the real axis and c is a given real point not contained in E. 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 work, 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.
| Original language | English |
|---|---|
| Journal | Journal of Approximation Theory |
| Volume | 65 |
| Issue number | 3 |
| Pages (from-to) | 261-272 |
| Number of pages | 12 |
| ISSN | 0021-9045 |
| DOIs | |
| Publication status | Published - 01.01.1991 |
Funding
* The work of the first author was supported by the German Research Association. The second author was supported by Cooperative Agreement NCC2-387 between the National Aeronautics and Space Administration and the Universities Space Research Association. This work was done while the authorsw ere visiting the ComputerS cienceD epartmento f StanfordU niversity.W e thank Gene Golub for his warm hospitality.
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