Noise Effects on Pade Approximants and Conformal Maps

TL;DR

This paper investigates how noise in data affects the accuracy of Pade and conformal map approximants for functions with branch points, revealing a universal scaling law for their breakdown.

Contribution

It introduces a universal scaling relation linking noise strength and approximation order, supported by theoretical proofs and physical examples.

Findings

Breakdown of approximants occurs at a predictable scale related to noise level.

The scaling law applies to functions with complex Riemann surfaces.

Physical models confirm the theoretical predictions.

Abstract

We analyze the properties of Pade and conformal map approximants for functions with branch points, in the situation where the expansion coefficients are only known with finite precision or are subject to noise. We prove that there is a universal scaling relation between the strength of the noise and the expansion order at which Pade or the conformal map breaks down. We illustrate this behavior with some physically relevant model test functions and with two non-trivial physical examples where the relevant Riemann surface has complicated structure