Spherical harmonic expansion of a power function

Let $$f$$ be an even continuous function on the sphere $$S^{n-1}$$.

Find a relation of the spherical harmonic expansion between coefficients of $$f^n$$ and those of $$f$$.

• perhaps you can give some more info? do you want a relation between the expansion coefficients of $f^n$ and those of $f$? – Carlo Beenakker Apr 28 at 15:24
• Would the answer $c_{l,i} = \langle Y_{l,i} | f^n \rangle$ satisfy you, and why not? – Michael Engelhardt Apr 28 at 15:51
• @Carlo Beenakker: Yes, I would like to find a coefficients of the expansion of $f^n$, in particular their relation with the expansion of $f$. Will add this to the question. Thank you. – user4164 Apr 28 at 16:06
• @Michael Engelhard: I would like to find coefficients of $f^n$ in terms of coefficients of $f$, i.e their relation. – user4164 Apr 28 at 16:13
• this would amount to an $n$-fold convolution of the expansion coefficients of $f$, I don't see how anything simpler would appear. – Carlo Beenakker Apr 28 at 17:06