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A generalization of the theorem of Hardy: A most general version of the uncertainty principle for Fourier integrals. (English) Zbl 0873.42005

If \(ab\geq 1/4\), and \(f(x)= O(R(x)\exp(-ax^2))\), \(\widehat f(x)=O(Q(x)\exp(-bx^2))\) as \(|x|\to\infty\), where \(\limsup_{|x|\to\infty}x^2\log|R(x)|=0\) (the same for \(Q(x)\)), then either \(ab=1/4\) or \(f=0\). Moreover, \(f(z)=\exp(-az^2)P(z)\), where \(P\) is an entire function of at most minimal type of order 2 (in fact, it cannot grow faster than \(R(z)\)); the same for \(\widehat f(z)\), with \(R(z)\) replaced by \(Q(z)\).

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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