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I'm interested if there is the explicit forms of basis functions in $L^2(S^n), n\geq 3$.

For $n=1, n=2$ basis functions are well known: $\{e^{ik\phi}\}_{k\in\mathbb{Z}}$, $\{p^{|m|}_n(\cos \gamma) e^{im\phi} | (\gamma,\phi)\in S^2, n \geq 0, m = \overline{-n,n}\}$. Also the following fact is known: \begin{equation} L^2(S^n) = \bigoplus_{k=0}^{\infty} H_k(S^n), \end{equation} where $H_k(S^n)$ - space of harmonic, homogeneous of degree $k$ polynomials on $S^n$. Actually, space $H_k(S^n)$ is obtained just by restricting $H_k(R^{n+1})$ to $S^n$: \begin{equation} \pi : H_k(R^{n+1})\rightarrow H_k(S^n): P(x)\mapsto P\left( \frac{x}{\|x\|} \right), \end{equation} which is also a bijection. As space $H_k(R^{n+1})$ is finite-dimensional, one can choose linearly independent system in $H(R^{n+1})$ and after mapping to sphere linear independence will preserve. So it is left to do the orthogonalization process and that's it for $H_k(S^n)$. This is a straight method to look for the exact basis, however it involves huge calculations.

Maybe someone did it before and there is a reference for the exact basis functions?

p.s. The question is motivated by calculating the integral: \begin{equation} I(\rho,\theta) = \int\limits_{S^n} Y_{k}^{i}(\omega) e^{i\rho (\omega,\theta)} \, d\omega, \end{equation} where $Y_{k}^i$ is a spherical harmonic function from basis in $H_k(S^n)$.

(most of the info I took from the book Stein&Weiss -- Introduction to Fourier Analysis in Euclidean spaces).

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    $\begingroup$ Folland, Harmonic analysis of the deRham complex on the sphere, doesn't quite give the answer, but he gives explicit functions giving a basis in each irreducible representation. So it remains to Gram-Schmidt these functions. $\endgroup$ – Ben McKay Oct 5 '16 at 16:03
  • $\begingroup$ Thank you for the answer, I will look for this paper. However I just realized that my question is a bit too general, since there are a lots of different basis on a sphere. What I look is more natural object: eigenfunctions of the Laplacian on the sphere (this way for example the basis on sphere in 3D is being built). $\endgroup$ – Fedor Goncharov Oct 6 '16 at 6:54
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    $\begingroup$ There are complicated but explicit formulas for an orthogonal basis of spherical harmonics of degree $l$, see the Wikipedia article. $\endgroup$ – Ivan Izmestiev Oct 12 '16 at 11:21
  • $\begingroup$ Thank you! That's I was looking for, too bad I forgot to check English wiki. $\endgroup$ – Fedor Goncharov Oct 13 '16 at 0:33
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Ivan Izmestiev answered my question, see comments below.

In addition, the question above inspired me to find the explicit result for such integrals (they arise in the framework of generalized Radon transforms):

https://hal.archives-ouvertes.fr/hal-01415990

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