We consider the orthogonalisation of the signature of a stochastic process as the analogue of orthogonal polynomials on path-space. Under an infinite radius of convergence assumption, we prove density of linear functions on the signature in $L^p$ functions on grouplike elements, making it possible to represent a square-integrable function on (rough) paths as an $L^2$-convergent series. By viewing the shuffle algebra as commutative polynomials on the free Lie algebra, we revisit much of the theory of classical orthogonal polynomials in several variables, such as the recurrence relation and Favard's theorem. Finally, we restrict our attention to the case of Brownian motion with and without drift, and prove that dimension-independent orthogonal signature exists with drift but not without. We end with numerical examples of how orthogonal signature polynomials of Brownian motion can be applied for the approximation of functions on paths sampled from the Wiener measure.

翻译：本文研究随机过程特征的正交化，将其视为路径空间上正交多项式的类比。在无穷收敛半径的假设下，我们证明了特征线性函数在类群元素上的$L^p$函数中的稠密性，从而可将（粗糙）路径上的平方可积函数表示为$L^2$收敛级数。通过将洗牌代数视为自由李代数上的交换多项式，我们重新审视了多变量经典正交多项式的大量理论，如递推关系与法瓦尔定理。最后，我们将研究聚焦于带漂移与无漂移的布朗运动情形，证明带漂移时存在与维度无关的正交特征，而无漂移时则不存在。文末通过数值算例展示了如何应用布朗运动的正交特征多项式来逼近从维纳测度采样的路径函数。