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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.

RBega2
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  • The spherical harmonics are the eigenfunctions of an Hermitian operator, right? If your Lipschitz function is in the same space, or even sometimes not, the eigenfunctions of an Hermitian operator are orthogonal and complete - that might be all you need. – Adrian Keister Jan 18 '19 at 20:47
  • Oh, wait: you need a finite combination. That's harder. Never mind. – Adrian Keister Jan 18 '19 at 20:48
  • I don't care so much about the finite combination, but more about the uniform Lipschitz bound (which does not readily follow from using the $L^2$ theory as far as I can tell). – RBega2 Jan 18 '19 at 20:50

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