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In the analysis of partial differential equations on Euclidean spaces, one of the most useful properties of the Fourier transform (and the related integral transforms) is that they take differentiation and turn it into multiplication, for example,

$ \frac{\partial^2 y}{\partial x^2} = f(x) $

becomes the following much simpler equation in the Fourier domain,

$ -\omega^2 \widehat{y} = \widehat{f}(\omega) $.

The underlying idea here is that differentiation can be understood to be a type of infinitesimal convolution with an element of the Lie algebra on $\Re^n$, and so by the convolution theorem it may be rewritten as a multiplication in the frequency domain. Similarly, for differential equations on spaces diffeomorphic to Lie groups (or those which happen to be faithful homogeneous spaces of a Lie group), one can use the same basic trick:

For example, on a sphere one can convert to spherical harmonics, which are really just the restriction of the coefficients for the matrix representations of $SO(3)$. The addition theorem then takes on the same role as the convolution theorem in the Euclidean case. An analogous argument can be made for cylinders, Bessel functions and the group $SE(2)$.

However, in practical applications (such as the types of mechanical engineering problems I am ostensibly employed to work on), most domains where PDEs are defined are not really equipped with some faithful action by a Lie group. But, is there perhaps a way to make this work anyway? For example, is it maybe reasonable to think of certain kinds of differential operators as maybe an infinitesimal monoid, or possibly categorical, convolution operator? This seems intuitive, but I am not really sure where to look for sources. It would be interesting if this perspective could be used to understand related Fourier-like transforms, for example the Chebyshev transforms.

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Note: one can define some "convolution-like" operators on manifolds or graphs. For example, some people take the mesh/graph Laplacian operator, and consider its eigenvectors to be the appropriate generalization of Fourier series. If this is true, then one could work backwards from the convolution theorem and define the convolution of two function to be the pointwise multiplication of the eigen functions. However, this forces all convolution to be commutative, which is not always desirable (for example in the case of the sphere...) –  Mikola Jun 4 '10 at 23:26
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