Is There a Generalized Fourier Transform for All Manifolds?

[redtree]

Is there a generalized form of the Fourier transform applicable to all manifolds, such that the Fourier transform in Euclidean space is a special case?

 

[Orodruin]

The corresponding thing in a (Riemannian or pseudo-Riemannian) manifold would be to look for expansions in terms of eigenfunctions of the operator $\nabla^2 = \nabla_a \nabla^a = g^{ab} \nabla_a \nabla_b$.