Let $f:\mathbb{S}^n\to \mathbb{R}$ be a Lipschitz function (i.e. so $$ \Vert f\Vert_{L}=\sup_{x\in \mathbb{S}^n} |f(x)|+\sup_{x\neq y\in \mathbb{S}^n} \frac{|f(x)-f(y)|}{d_{\mathbb{S}^n}(x,y)}<\infty. $$
Can one find a sequence $S_N(f)$ consisting of the linear combination of $N$ spherical harmonics so that $$ \Vert f-S_N(f)\Vert_0=\sup_{x\in \mathbb{S}^n} |f(x)-S_N(f)(x)|\to 0 $$ and so for $N$ sufficiently large $$ \Vert S_N(f)\Vert_L\leq 2 \Vert f\Vert_L. $$
In other words, can $f$ be uniformly approximated by a finite linear combination of sphereical harmonics whose Lipschitz norm is uniformly bounded (by twice the Lipschitz norm of $f$ but this is not so important). I believe this can be shown when $n=1$ by using the Fejer Kernel.