Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ is smooth and analytic on every line). The claim is that $f$ is itself real analytic on all of $\mathbb R^p$.

I do have my own proof of this fact (rough sketch: use Baire category theorem to show that there is a open set of directions $U$ in $S^{p-1}$ on which there are $A,B \geq 0$ such that $$|g_d^{(n)}(0)| \leq AB^nn!$$ for all $n$ and $d \in U$. Use linear algebra and a smart choice of directions to deduce that there exist $C,D \geq 0$ such that $$\left|\frac{\partial^n}{\partial x_1^{n_1}\cdots\partial x_p^{n_p}}f(0)\right| \leq CD^nn!$$ for all $(n_1,\ldots,n_p) \in \mathbb N^p$, and $n = n_1+\cdots+n_p$. Then there is a locally convergent power series with the same derivatives as $f$ at $0$. Apply analyticity on each line to conclude that $f$ is this power series. Repeat argument with arbitrary $x$ instead of $0$.), but my supervisor is convinced that this is actually a classical result.

However, despite my attempts, I cannot find a reference for it or anything similar. Does anyone know of one?

  • 1
    $\begingroup$ This looks helpful: books.google.com/… $\endgroup$ – Jonas Meyer Jul 29 '14 at 3:57
  • $\begingroup$ (In case the link doesn't work for everyone, it is to page 152 of Real submanifolds in complex space and their mappings by Baouendi, Ebenfelt, Rothschild.) $\endgroup$ – Jonas Meyer Jul 29 '14 at 4:03
  • $\begingroup$ Thank you very much! It's also nice to know that half of my proof is essentially the same. $\endgroup$ – E.Y.Jaffe Jul 29 '14 at 4:22

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