0
$\begingroup$

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 $f\in L^2(S^1)$

$$f=\sum_{n}c_n\phi_n.$$

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 $M=S^1$?

$\endgroup$
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.