Abstract
We prove uncertainty principles of Hardy type that limit the possibility to localize a distribution and its Fourier transform near the cone \(q=0\), for a non-degenerate quadratic form \(q\) of arbitrary signature. The results we present are well known for positive definite \(q\). We describe two types of distributions that optimize the uncertainty principle in this case. The first type are the distributions \(f\) such that \(f\) and \(\widehat{f}\) are supported on the cone \(q=0\). The second type are distributions that depend only on \(q\), that are essentially their own Fourier transform, and that decay like Gaussian functions as \(|q|\rightarrow \infty \).
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Communicated by Karlheinz Gröchenig.
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Demange, B. Uncertainty Principles and Light Cones. J Fourier Anal Appl 21, 1199–1250 (2015). https://doi.org/10.1007/s00041-015-9401-6
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DOI: https://doi.org/10.1007/s00041-015-9401-6