# Johannes Kepler Symposium on Mathematics

As part of the Johannes Kepler symposium on mathematics Klaus Schiefermayr, Mathematik und Statistik, Fakultät für Technik und Umweltwissenschaften, FH OÖ Studienbetriebs GmbH, will give a public talk (followed by a discussion) on Wed, Nov. 3, 2010 at 16:15 o'clock at HS 13 on the topic of "Inequalities for the minimum deviation of Chebyshev polynomials and inverse polynomial images" . The organziers of the symposium,

O.Univ.-Prof. Dr. Ulrich Langer,
Univ.-Prof. Dr. Gerhard Larcher
A.Univ.-Prof. Dr. Jürgen Maaß, and
die ÖMG (Österreichische Mathematische Gesellschaft),

hereby cordially invite you.

Series B - Mathematical Colloquium:

The intention is to present new mathematical results for an audience interested in general mathematics.

### Inequalities for the minimum deviation of Chebyshev polynomials and inverse polynomial images

“Chebyshev polynomials are everywhere dense in numerical analysis.” This remark has been attributed to a number of distinguished mathematicians and numerical analysts. The classical Chebyshev polynomials of the first kind, which is attributed in the above quotation, is typical for the set of one interval: Especially it is the fact that this polynomial is the minimal polynomial on the interval $[-1,1]$ with respect to the supremum norm, which involves so many applications in different areas of mathematics, most of them in numerical analysis. In this talk, we present some generalizations of these Chebyshev polynomials of the first kind. More precisely, we consider the minimal polynomial with respect to the supremum norm on a general set S of the complex plane, especially on the union of several real intervals. Again, such a polynomial is called the Chebyshev polynomial on S. In this context, the notion of an inverse image of $[-1,1]$ of a given polynomial mapping is crucial, since each polynomial is (suitably normed) the Chebyshev polynomial on its inverse image.