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 $f$.
$\begingroup$
$\endgroup$
7
-
1$\begingroup$ 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 BeenakkerCommented Apr 28, 2021 at 15:24
-
2$\begingroup$ Would the answer $c_{l,i} = \langle Y_{l,i} | f^n \rangle $ satisfy you, and why not? $\endgroup$– Michael EngelhardtCommented Apr 28, 2021 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. $\endgroup$– user4164Commented Apr 28, 2021 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. $\endgroup$– user4164Commented Apr 28, 2021 at 16:13
-
3$\begingroup$ 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 BeenakkerCommented Apr 28, 2021 at 17:06
|
Show 2 more comments