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Let $S^n$ be the $n$-sphere. It is well-known that $S^n$ is the one-point compactification of $\mathbb{R}^n$ via the stereogarphic projection.

Moreover, the Schwartz space on $\mathbb{R}^n$ may be identified with the space of smooth functions on $S^n$ whose derivatives of all orders vanish at a fixed point on $S^n$, as in this ME post.

Now, my question is

Is there any description of a general smooth function on $S^n$ as a function on $\mathbb{R}^n$?

In other words, I am looking for some identification of $C^\infty(S^n)$ with a function space on $\mathbb{R}^n$.

My guess is that such a function must have some asymptotics of all orders of derivative at infinity, but I cannot find a precise answer. Could anyone help me?

Keith
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    Could you precise the operations that must be kept under this identification? Addition, multiplication, derivation? – Christophe Boilley Mar 16 '25 at 09:10
  • @ChristopheBoilley Indeed, I think that the identification must preserve addition and multiplication. I think that derivatives must be preserved too, but I do not see how preservation of derivatives should look like. – Keith Mar 16 '25 at 16:36

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