General approach to function approximation

TL;DR

This paper presents a general framework for function approximation using functionals, unifies existing methods, and introduces new expansion types including rational approximations.

Contribution

It offers a unified interpretation of known approximations and proposes several novel expansion methods, notably three rational approximation techniques.

Findings

Unified interpretation of existing approximation methods

Introduction of new expansion types including rational approximations

Framework applicable to a wide range of function approximation techniques

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.