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Let $\mathcal{H}_k\subset L^2(S^n)$ be the space of sphere harmonics of degree $k$, i.e., they are the eigenfunctions of $\Delta_{S^n}$. And let $\Pi_k:L^2(S^n)\to \mathcal{H}_k$ be the orthogonal projection operator. It has a kernel defined by $$ \Pi_kf(x)=\int_{S^n}\Pi_k(x,y)f(y)dS(y) $$ A Zonal spherical harmonics $Y_0^{k}(x)$ is defined as $$ Y_0^{k}(x)=\frac{\Pi_k(x,y_0)}{\sqrt{dim \mathcal{H}_k}} $$ here, $y_0$equal to the north pole. From the definition, we can immediately get $$ \|Y_0^{k}(x)\|_\infty\leq C\sqrt{dim \mathcal{H}_k}\leq Ck^{\frac{n-1}{2}} $$ But I also want to get the $L^p$ norms, so I want to know how they look like, then I can compute.

Thanks for any reference.

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    $\begingroup$ I would guess that the zonal spherical harmonics are the only spherical harmonics with the property that the value of the function only depends on the $n$th coordinate. So in that sense, they are functions of one variable only. Maybe that helps? $\endgroup$ Feb 11, 2014 at 7:10
  • $\begingroup$ Is there an explicit formula for the $L^p$ norms of zonal spherical harmonics? The paper faculty.fiu.edu/~decarlil/Preprints/Proc4.pdf proves estimates for them, which suggests that no such formula is known. $\endgroup$
    – Henry Cohn
    Feb 11, 2014 at 16:24
  • $\begingroup$ I am also not aware of any explicit formula for the $L^p$-norms, but in general one can probably obtain estimates for them by means of certain Bernstein type inequalities for the Gegenbauer polynomials. Perhaps the following article can also be of interest: [N.J. Kalton and L. Tzafriri, The behaviour of Legendre and ultraspherical polynomials in Lp-spaces, Canad. J. Math. 50 (1998), 1236-1252.] $\endgroup$ Feb 11, 2014 at 19:29

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Since $S^n \cong \mathrm{SO}(n+1)/\mathrm{SO}(n)$, the zonal spherical harmonics you are asking for arise as the spherical functions for the compact Gelfand pair $(\mathrm{SO}(n+1),\mathrm{SO}(n))$, which are known to be the Gegenbauer (also called ultra-spherical) polynomials.

A good reference, including background on Gelfand pairs, is (Section 7.2 of) the recent book [G. van Dijk, Introduction to Harmonic Analysis and Generalized Gelfand Pairs, de Gruyter, Berlin, 2009], but it can also be found in other books, e.g., Analysis on Lie Groups by Faraut.

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