## 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

### Keywords:

theorem of Hardy; uncertainty principle
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### References:

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