The Heisenberg principle and positive functions (Q1958970)

The authors consider Fourier couples \((f,\hat{f})\) on \({\mathbb R}^d\) satisfying the conditions: \(f\), \(\hat{f}\) are real, even an not identical zero; \(f(0)\leq 0\) and \(\hat{f}\leq 0\); \(f(x)\geq 0\) for \(\parallel x\parallel\geq a_f\), \(\hat{f}(y)\geq 0\) for \(\parallel y\parallel\geq a_{\hat{f}}\). Let \[ A(f)=\inf\{r>0:f(x)\geq 0,\parallel x\parallel>r\} \] and \[ B_d=\inf A(f)A(\hat{f}). \] The authors prove that \[ B_d\geq \frac{1}{\pi}\Big(\frac 12\Gamma\Big(\frac d2+1\Big)\Big)^{\frac 2d}>\frac{d}{2\pi e}. \] Let \({\mathcal B}_d\) be the constant arising considering only functions in the Schwartz space \({\mathcal S}({\mathbb R}^d)\). Then \[ B_d\leq {\mathcal B}_d\leq\frac{d+2}{2\pi},B_d\geq \frac 12{\mathcal B}_d. \] Suppose that there exists a number field \(F\) of degree \(d\) and discriminant \(D\) such that \(\zeta_F(s)\) has a zero in \((0,1)\). Then \[ {\mathcal B}_d\geq d|D|^{-\frac 1d}. \] If \(d\) is a multiple of \(48\), then \({\mathcal B}_d>0 \).