When one expands a function defined on the circle $S^1$ into its Fourier series what is really being done is to consider the expansion on the complete set of function $\{\phi_n\}$ with $\phi_n(x) = e^{in x}$, so that in the end we have for $f\in L^2(S^1)$


The same can also be done in the sphere $S^2$. There we have the spherical harmonics $Y^m_l$ and they can be seen as just generalizing the Fourier series for the two dimensional case of the sphere. In that case $\{Y^m_l\}$ gives a complete set of functions on the sphere and we can expand $f\in L^2(S^2)$

$$f=\sum_{l=0}^\infty\sum_{m=-l}^l c_{ml}Y^m_l.$$

Now these are not just any complete sets of function, they are rather "special".

My question is: given a smooth manifold $M$, is there some known construction that yields a complete set of functions $\{\psi_n\}$ so that we can write for any $f\in L^2(M)$

$$f=\sum_{n}c_n \psi_n,$$

and so that the construction reduces to the spherical harmonics and to the Fourier series when $M=S^2$ and $M=S^1$?


1 Answer 1


Yes, the appropriate functions are the eigenfunctions for the Laplace-Beltrami operator. On any compact manifold $M$ with a Riemannian metric, the Laplace-Beltrami operator acting on functions has discrete eigenvalues, and there is a complete set of eigenfunctions such that every $L^2$ function has an expansion as a sum of eigenfunctions, convergent in the $L^2$ norm. When you specialize to either $S^1$ or $S^2$ with the metrics induced from their standard embeddings in Euclidean space, you get the functions you mentioned.


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