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I am trying to find an explicit example of smooth real-valued compactly supported function $u$ in $\mathbb R^n$, $n \geq 2$, such that its Fourier transform $\widehat u$ does not have zeros. By the Paley-Wiener characterization theorem it is sufficient to find an entire function $U$ on $\mathbb C^n$ such that:

  1. for any $N$ there exists $C_N$ such that $$ |U(z)| \leq C_N (1+|z|)^{-N} e^{B|\mathop{\mathrm{Im}} z|} $$ for some $B > 0$;
  2. $U(z) \neq 0$ for $z \in \mathbb R^n$.
  3. $U(z) = \overline{U(-z)}$, $z \in \mathbb R^n$.

On the other hand, by the Wiener $L^1$-approximation theorem it is sufficient to find a smooth compactly real-valued function $u$ in $\mathbb R^n$ such that the span of its shifts $u_a = u(\cdot-a)$, $a \in \mathbb R^n$, is dense in $L^1(\mathbb R^n)$. Please, help me.

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Let's start with $n=1$. Let $\phi$ be any even compactly supported smooth function. Then $u(x)=(\phi\ast\phi)(x)\,e^{-x^2}$ is smooth, has compact support, and its Fourier transform is strictly positive. In $\mathbb{R}^n$ take $u(x_1)\dots u(x_n)$.

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  • $\begingroup$ ok that works, $U(\omega) = |\Phi(\omega)|^2 \ast (b e^{- a \omega^2})$ well done $\endgroup$ – reuns Feb 13 '16 at 20:46

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