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).



Basis on the sphere in multidimensions
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>


