Operators That Preserve x2 and Associated Summability Methods.
Jerry P. King
Lehigh University
Many classical positive linear operators connect three areas of mathematics:
approximation theory, probability theory, and summability theory. A key role is played
by the functions e0(x) = 1, e1(x) = x, and e2(x) = x2. Most of the classical operators
preserve both e0 and e1. In 2003 King published an example of a non-trivial sequence of
positive linear operators, Ln, with the property that Ln(e0) = e0 and Ln(e2) = e2.
Subsequently, a number of papers have appeared which generalize both the method used
by King and the results he obtained.
After a brief introduction to the connections among approximation, probability, and
summability, the talk outlines some recent and new results in this area. Emphasis is
given to summability methods associated with the operators and to opportunities for
future research.