turkmath.org

This talk deals with historical overview and survey on weighted approximations starting from the contributions of Prof. Akif Gadjiev (1937-2015) to the most recent results presents by me and the other researchers. The weighted approximation methods have been developed for around 40 years and it is an active research areas for many approximation processes with different aspects, techniques and approaches.

One of the fundamental problems of approximation theory is the existence of a sequence of polynomials which lie above (or below) of a given function and approximate it arbitrarily well. We present an overview for the problem from $C(R)$ to the $L_p$-spaces and its generalizations. We also discuss some open problems and describe possible directions of future researches in the topic.