We introduce a framework for constructing orthogonal polynomials on path space. Beginning with an introduction to signatures which play the role of polynomials, and we orthogonalise these features to obtain \(L^2\)-convergent series for square-integrable path functionals. Under an infinite radius of convergence assumption, we prove linear functionals on the signature are dense in \(L^p\).

Identifying the shuffle algebra with polynomials over the free Lie algebra, we generalise orthogonal polynomial theory: establishing recurrence relations, a Favard-type theorem, and connections to spectral measures. For Brownian motion, a natural (dimension-independent) orthogonal basis exists only with time-augmentation, yielding explicit Itô-orthogonal polynomials.

In ongoing work with Markus Reiß and Christian Bayer, we apply these methods to classify Ornstein-Uhlenbeck processes, obtaining closed-form expected signatures and optimal discriminative features for hypothesis testing