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Someone told me that every continuous function on $\mathbb{S}^2$ could be expressed as a uniform limit of restrictions to $\mathbb{S}^2$ of polynomials. Does this result come from the Stone-Weierstrass theorem? Could anyone be able to explain to me what it means formally?

Thanks for your help!

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Just not to worry that the restriction of polynomials will satisfy the hypotheses of Stone-Weierstrass, let's use Tietze's theorem to extend our function from $S^2$ to the cube. Now we can apply Stone-Weierstrass to this continuous extended function, approximate it by polynomials, and restrict them.

Aloizio Macedo
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    we can consider instead $F(r, \theta,\varphi) = r , G(\theta,\varphi)$ where $G(\theta,\varphi)$ is the function on the sphere we'd like to approximate, and $F$ is a continuous function on the ball $|x| = r < R$ ? – reuns Jun 23 '16 at 18:53
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    and see his other question which is interesting http://math.stackexchange.com/questions/1836490/spherical-harmonics-beltrami-operator#1836490 – reuns Jun 23 '16 at 18:56