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The statement which is not very clear is this: if $\phi \in C^{\infty}_0$ then its Fourier transform extends to a complex-analytic function.

Of course the candidate extension is $\hat{\phi}(z)=\int e^{-i<x,\xi+i\eta>}\phi(x) dx$ where $x,\xi,\eta$ are in $\mathbb{R}^n$ and $z=\xi+i\eta$. First of all this integral is absolutely convergent since $\phi$ has compact support, and this is ok. Then we prove estimates on the derivatives: $$\partial_{z_j}\hat{\phi}(z)= \int e^{-i<x,z>}(-i)x_j\phi(x)dx$$ and supposing the support of $\phi$ included in the ball centered at the origin with radius $A$ we can prove the estimate $$|\partial_{z_j}\hat{\phi}(z)|\leq c_je^{A |Im(z)|}.$$ And analogously

$$|\partial^{\alpha}_{z}\hat{\phi}(z)|\leq c_{\alpha}e^{A |Im(z)|}$$ where $\alpha $ is a multindex. I don't understand how these estimates prove the analyticity of the function $\hat{\phi}(z)$. Shouldn't we prove estimates of the type $|\partial^{\alpha}f| \leq |\alpha !|C^{|\alpha|}$?

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It's a general fact that if $f(z,t)$ is analytic in $z$ for every $t$ and the integral $F(z)=\int_X f(z,t)\,d\mu(t)$ converges locally uniformly wrt $z$, then $F$ is analytic. Probably the easiest way to show this is to use Morera's theorem. It is generally more pleasant to integrate an integral than to differentiate it.

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Thank you. What is the variabile $t$ in this example? Isn't there any way to compute the classical estimates (if they are correct also in several complex variable)? – balestrav Jun 27 '12 at 20:03
@balestrav $t$ can be any variable of integration for an arbitrary measure space. Higher dimensions should not be a problem: as far as I remember, a separately analytic function is analytic. – user31373 Jun 27 '12 at 20:16

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