Fourier series
Riemannian manifolds

If the domain is not a group, then there is no intrinsically defined convolution. However, if X {\displaystyle X} is a compact Riemannian manifold, it has a Laplace–Beltrami operator. The Laplace–Beltrami operator is the differential operator that corresponds to Laplace operator for the Riemannian manifold X {\displaystyle X}. Then, by analogy, one can consider heat equations on X {\displaystyle X}. Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace–Beltrami operator as a basis. This generalizes Fourier series to spaces of the type L 2 (X) {\displaystyle L^{2}(X)}, where X {\displaystyle X} is a Riemannian manifold. The Fourier series converges in ways similar to the {\displaystyle } case. A typical example is to take X {\displaystyle X} to be the sphere with the usual metric, in which case the Fourier basis consists of spherical harmonics.
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