Approximating functions on ${\mathbb R}^+$ by exponential sums

TL;DR

This paper introduces a novel method for approximating functions on the positive real line using exponential sums, leveraging multi-point Padé approximation and continued fractions for high accuracy.

Contribution

The paper develops a new approximation technique combining multi-point Padé approximation of the Laplace transform with continued fractions, enhancing function approximation on ${ m f R}^+$.

Findings

Accurately approximates various functions including Gaussian and probability density functions.

Demonstrates high precision in approximating special functions like gamma and Barnes G-functions.

Provides examples showing the method's effectiveness across different function types.

Abstract

We present a new method for approximating real-valued functions on ${\mathbb R}^+$ by linear combinations of exponential functions with complex coefficients. The approach is based on a multi-point Pad\'e approximation of the Laplace transform and employs a highly efficient continued fraction technique to construct the corresponding rational approximant. We demonstrate the accuracy of this method through a variety of examples, including the Gaussian function, probability density functions of the lognormal and Gompertz-Makeham distributions, the hockey stick and unit step functions, as well as a function arising in the approximation of the gamma and Barnes $G$-functions.