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Let $f \in C_c^\infty(\mathbb{R}^n)$. Then, $$\hat{f}(\xi)= (2\pi)^{\frac{-n}{2}}\int_{\mathbb{R^n}}\exp(-ix\xi)f(x){d}x$$

can be (i) analytically continued to an entire function and (ii) for $r>0$ there holds: $\hat{f}(\cdot + ib)$ is uniformly rapidly decreasing $\forall\; b\in B(0,r)$.

Can somebody help me to see (i) and (ii)?


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up vote 1 down vote accepted

(i) The integral makes sense for every $\xi\in\mathbb C$, since you are integrating a continuous function over a compact set. Since $\xi\mapsto \exp(-ix\xi)$ is holomorphic in $\mathbb C$, so is the integral. (Holomorphicity is preserved under all kinds of limits, including integration, which is also a kind of a limit.)

(ii) If you integrate by parts, throwing derivative onto $f$, you get $\xi$ in denominator. You can do this as many times as you want.

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