Computation of some transcendental integrals from path signatures

TL;DR

This paper demonstrates how to compute certain transcendental integrals along paths of finite variation using path signatures, enabling the calculation of these integrals and related topological invariants from signature data.

Contribution

It introduces a method to compute transcendental integrals from path signatures, extending the applicability of signatures to complex analysis and topology.

Findings

Integral functions are entire holomorphic under specified conditions.

Values of the integrals can be reconstructed from integer samples.

Signature-based computation of winding numbers around submanifolds is possible.

Abstract

It is shown that if $\gamma$ is a path of finite $p$ variation ($1\leq p< 2$) in a euclidean vector space and $f,g,h$ are Lipschitz functions on the trace of $\gamma$ then $s\mapsto F(s)=\int_\gamma f^sg dh$ defines an entire holomorphic function provided the convex hull of the image of $f$ does not contain zero. If in addition $|\log z|\leq \log 2$ on the convex hull of the image of $f$ then for any $s\in \mathbf{C}$, $F(s)$ can be computed from the nonnegative integer values $\{F(k)\}_{k\in \mathbf{N}}$. If in addition to these hypotheses each of $f,g,h$ is a polynomial, then the values $F(k)$ are computable directly from the signature of $\gamma$ thus all values of $F(s)$ are computable from the signature. As a special case the winding number of a closed path $\gamma$ around an affine submanifold of codimension two is computed from finitely many terms of the signature provided certain estimates are satisfied.