The space $LV$ of free loops on a manifold $V$ inherits an action of the circle group $\T$. When $V$ has an almost-complex structure, the tangent bundle of the free loopspace, pulled back to a certain infinite cyclic cover $\LV$, has an equivariant decomposition as a completion of $\bT V \otimes (\oplus \C(k))$, where $\bT V$ is an equivariant bundle on the cover, nonequivariantly isomorphic to the pullback of $TV$ along evaluation at the basepoint (and $\oplus \C(k)$ denotes an algebra of Laurent polynomials). On a flat manifold, this analogue of Fourier analysis is classical.
"The tangent bundle of an almost-complex free loopspace." Homology Homotopy Appl. 3 (2) 407 - 415, 2001.