Signature inversion of $C^1-$axial linear curves

TL;DR

This paper presents a method to invert signatures of $C^1$-axial linear curves, enabling the recovery of derivatives at any point and providing convergence estimates and continuity properties of the inverse signature map.

Contribution

It introduces a signature inversion scheme for $C^1$-axial linear curves, including convergence rates and continuity results for the inverse signature.

Findings

Derivatives can be recovered from signature coefficients with $rac{k}{k+l} o x$.

Provides quantitative convergence rate estimates.

Establishes a modulus of continuity for the signature inverse.

Abstract

We introduce a signature inversion scheme for $C^1$-axial linear curves which are widely used in various areas. We show that in the presence of a linear coordinate function, the derivatives of the underlying curve at any point $x$ can be recovered by tracking the signature coefficients $S_{k,l}$ with $\frac{k}{k+l} \to x$. We furthermore give a quantitative estimates for the convergence rate in this inversion scheme and establish a modulus of continuity of the signature inverse $S^{-1}$ under different topologies by using this inversion procedure.