General approach to function approximation

Abstract

Having a function $f$ and a set of functionals $\{{\mathcal{C}}_n\}$, $c_n^f\equiv {\mathcal{C}}_n\left(f\right)$, one can interpret function approximation very generally as a construction of some function ${\mathcal{A}}_N^f$ such that $c_n^f={\mathcal{C}}_n\left({\mathcal{A}}_N^f\right)$. All known approximations can be interpreted in this way and we review some of them. In addition, we construct several new expansion types including three rational approximations.