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A survey on uncertainty principles related to quadratic forms. (English) Zbl 1107.30021

Various distribution versions of the uncertainty principles of Hardy, Beurling and Morgan are given. In particular, Hardy’s uncertainty principle is proved for temperate distributions \(f\) on \(R^d\) such that \(e^{\pm\pi q}f\) and \(e^{\pm\pi q'}\widehat f\) also tempered, when \(q\) and \(q'\) are two positive definite quadratic forms on \(R^d\). The case of general nondegenerate forms is also considered. A special attention is given to the case \(d=2\) and \(q(x,y)= q'(x,y)=2xy\). In this case all tempered distributions \(f\) satisfying the above hypotheses are completely described they are obtained from the Gaussian function \(e^{-\pi tx^2-\pi/ty^2}\) through a small number of operations.

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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