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 $f$.

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

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