Approximating functions on ${\mathbb R}^+$ by exponential sums
This work provides a new approximation tool for functions on ℝ⁺, relevant to fields like signal processing and probability, but the results are demonstrated on specific examples without broad SOTA claims.
The paper introduces a method for approximating real-valued functions on the positive real line using exponential sums, based on multi-point Padé approximation of the Laplace transform and continued fractions. The method achieves high accuracy across diverse examples, including Gaussian, lognormal, and step functions.
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 Padé 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.