Let $f$ be a continuous function $[-1,1]\to\mathbb{R}$. Consider an integral operator $A$ on the unit sphere $S^{d-1}$ of $\mathbb{R}^d$, which acts on $\phi\in\mathcal{L}^2(S^d)$ as $$A\,\phi(x) = \int_{S^d}f(x\cdot y)\phi(y)dy\,.$$ By Mercer's theorem $$A(x,y) = \sum_{k\in\mathbb{N}} \lambda_k\,\psi_k(x)\psi_k(y)\,.$$ Can I claim that the functions $\psi$'s are all spherical harmonics?
I know this is the case if $f$ is analytical on $(-1,1)$, but is the continuity enough? Most of results I could find claim that $A$ is a zonal kernel and so has such a decomposition in spherical harmonics. But I couldn't fine a clear definition of zonal kernel. Most of references just ask that $A(x,y) = f(x\cdot y)$, giving no regularity request on $f$, but I guess some sort of regularity is actually implied.