  We develop a Fourier approach to rough path integration, based on the series
decomposition of continuous functions in terms of Schauder functions. Our
approach is rather elementary, the main ingredient being a simple commutator
estimate, and it leads to recursive algorithms for the calculation of pathwise
stochastic integrals, both of It\^o and of Stratonovich type. We apply it to
solve stochastic differential equations in a pathwise manner.
