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Assume that a function $f: R \rightarrow R$ is smooth and even. Does there exist a smooth function $g:R \rightarrow R$ such that $f(x)=g(x^2)$ ?

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3 Answers 3

Incidentally, this follows easily from a simple integral formula for the derivatives of $g$ (see this question that I recently posted). Indeed, the Taylor formal power series in $0$ of an even $C^{\infty}$ function $f$ is of course in $\mathbb{R}[[x^2]]\, .$

(more details). Such a function $g$ necessarily has to be defined as $g(x):=f(\sqrt x)$ for $x > 0$. This makes it $C^\infty$ on $\mathbb{R_+}$ as a composition of smooth functions. However, it is not immediately obvious from the composition formula that the derivatives for $x > 0$, $\, g^{(k)}(x),$ do have a limit for $x\to 0$, which is of course a necessary condition for $g$ to be extendable to a $C^\infty$ function on $\mathbb{R}$. But this is clear from the representation

$$\frac{g^{(k)}}{k!} (x^2)=(2x)^{-2k+1}k {2k \choose k}\, \int_0^x (x^2-t^2)^{k-1}\frac{f^{(2k)}}{(2k)!}(t) dt\, $$

that exhibits the $k$-th Taylor coefficient of $g$ as an integral mean of the $2k$-th Taylor coefficients of $f$ on the interval on $[0,x]$, so that $$\lim_{x > 0\atop x \to 0} \frac{g^{(k)}}{k!} (x) = \frac{f^{(2k)}}{(2k)!}(0)\, .$$ In general, for a function $g\in C^\infty(\mathbb{R_+})$, all derivatives being continuously extendable at $0$ is also sufficient condition for $g$ to be smoothly extendable on $\mathbb{R}\, $ (as an easy instance of the Whitney extension theorem; or by Borel's theorem, extending $g$ on the left half-line by any $h\in C^\infty(\mathbb{R})$ with prescribed derivatives at $0$, or by more elementary arguments ad hoc).

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1  
(on the integral formula: by density, it suffices to prove it for polynomials, and by linearity, it suffices to consider $f(x)=x^n$. Substituting $s:=t^2/x^2$ in the integral, the identity reduces to the classical relation between the Beta and the Gamma functions). –  Pietro Majer Aug 10 '11 at 7:56

Yes, this is a theorem of Hassler Whitney:

Whitney, Hassler Differentiable even functions. Duke Math. J. 10, (1943). 159–160.

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See the answer (here) for more information: Smooth functions which are invariant under a compact Lie group representation factor smoothly over the basic invariant polynomials.

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