Recent Developments in the Theory of Polynomials
by
Kum Kum Dewan
JAMIA MILLIA ISLAMIA UNIVERSITY, NEW DELHI, INDIA

In the early part of this century, de la Vallie Poussin raised the following question of best approximation: Is it possible to approximate every polygonal line by polynomials of degree n with an error of o(1/n) as n becomes large? This question was answered in the negative by S. Bernstein. For this, he proved and made considerable use of an inequality concerning the derivatives of polynomials. This inequality was the starting point of a considerable literature in polynomial approximation theory. In fact inequalities of Markov and Bernstein-type are fundamental for the proof of many inverse problems in approximation theory and frequently further research in inverse theorems have depended upon first obtaining their corresponding generalization or analogue. In this talk we will discuss some of the investigations which have centered about these inequalities. Some new results will also be presented.

Date received: August 31, 2001