The Heisenberg principle and positive functions. (Principe d’Heisenberg et fonctions positives.) (French. English summary) Zbl 1298.11105
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 \({\mathcal B}_d>0 \).
Reviewer: Florin Nicolae (Berlin)
MSC:
11R42 | Zeta functions and \(L\)-functions of number fields |
42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
42B10 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
References:
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