An overview of approximation methods in function spaces, part I

In this presentation, we explore polynomial approximation schemes within function spaces. While Taylor polynomials are fundamental in polynomial approximation theory, there are instances where they may not be the most suitable candidates. Without entering into technical details, we will discuss some summation methods, with a particular emphasis on the well-known Cesaro means. Our focus remains primarily on Hardy and Dirichlet spaces, although other function spaces also make appearances in the discussion. Moreover, within the broader context of super-harmonically weighted Dirichlet spaces, we establish that Fejer polynomials and de la Vallee Poussin polynomials serve as appropriate approximation schemes.

This work has evolved over an extended period and is the result of collaborative efforts with O. El-Fallah, E. Fricain, K. Kellay, H. Klaja, M. Nasri, P. Parisé, M. Shirazi, W. Verreault, T. Ransford, and M. Withanachchi in various combinations.