A simple alternative in approximation and asymptotic expansion by exponential/trigonometric functions

Abstract

Function approximation plays a crucial role in applied mathematics and mathematical physics, involving tasks such as interpolation, extrapolation, and studying asymptotic properties. Over the past two centuries, several approximation methods have been developed, but no universal solution has emerged. Each method has its own strengths and weaknesses. The most commonly used approach, rational Padé approximants, has limitations, performing well only for arguments x < 1 and often containing spurious poles. This report introduces a new and straightforward procedure for exponential/trigonometric approximation that addresses these limitations. The method demonstrates accurate fitting capabilities for various functions and solutions of second-order ordinary differential equations, whether linear or nonlinear. Moreover, it surpasses the performance of Padé approximants. Notably, the proposed algorithm is remarkably simple, requiring only four values of approximating functions. The provided examples show case the potential of this method to offer a straightforward and reliable approach for a wide range of tasks in applied mathematics and mathematical physics.