Approximation systems

TL;DR

This paper introduces approximation systems as a broad generalization of Taylor approximations, providing theoretical foundations, error bounds, convergence criteria, and practical implementation insights.

Contribution

It develops the theory of approximation systems, extending Taylor approximation concepts with new error and convergence analysis, and discusses numerical implementation.

Findings

Established error bounds for approximation systems

Provided convergence criteria for these systems

Demonstrated practical numerical implementation

Abstract

We introduce the notion of an approximation system as a generalization of Taylor approximation, and we give some first examples. Next we develop the general theory, including error bounds and a sufficient criterion for convergence. More examples follow. We conclude the article with a description of numerical implementation and directions for future research. Prerequisites are mostly elementary complex analysis.