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I want to find a Fourier pair $(r,\hat{r})$ satisfying some conditions listed below and making $\hat{r}(0)$ as small as possible. The requirements for $(r,\hat{r})$:

  1. $r,\hat{r}\in L^1(\mathbb{R})$, defined by the relation \begin{align*} \hat{r}(\alpha)=\int_{-\infty}^{\infty}r(u)e(-\alpha u)du,\qquad e(u)=e^{2\pi iu}. \end{align*}
  2. $r \geq 0$ on $\mathbb{R}$ and $r(0)=1$.

  3. $\hat{r}=0$ outside the interval $[-1,1]$.

For example, if I choose $r(u)=\left({\sin\pi u\over \pi u}\right)^2$, then $\hat{r}(\alpha)=\max(1-|\alpha|,0)$, the pair meets our conditions, and $\hat{r}(0)=1$.

What is the smallest value of $\hat{r}(0)$, and which pair gives this value?

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