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It is well know that the spherical harmonics $Y_{l,m}(\theta,\varphi)$ are basis for $L_2(S^2)$ ( All square integrable functions on the unit sphere ) .

What about the basis of the space of all square integrable functions on the sphere of radius R ? Can we say this basis is the set of the same spherical harmonics of the case of the unit sphere ?

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  • $\begingroup$ I think yes. It is just a result of dilatation $\endgroup$
    – Kroki
    Aug 2, 2017 at 8:07

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In the space $L^2(S^1)$, we already have the harmonics (orthogonal basis) of the Fourier series of functions of any period $T$:

$$e_n(x)=(\sqrt T)^{-1}e^{2\pi inx/T}$$

If $T\to\infty$, then the Fourier series will become the Fourier transform. I wouldn't expect the spherical harmonics be different to the Fourier series, though I don't have the exact formula. Perhaps there will be a scale factor in the coefficient and the exponential.

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