Nazarov's uncertainty principles in higher dimension

Details Nazarov's uncertainty principles in higher dimension Nazarov's uncertainty principles in higher dimension

2006-12-13

arxiv.org/abs/math/0612367v1

Journal of Approximation Theory (04/05/2007) doi:10.1016/j.jat.2007.04.005

Mathematics

Classical Analysis and ODEs

Scientific paper

10.1016/j.jat.2007.04.005

In this paper we prove that there exists a constant $C$ such that, if $S,\Sigma$ are subsets of $\R^d$ of finite measure, then for every function $f\in L^2(\R^d)$, $$\int_{\R^d}|f(x)|^2 dx \leq C e^{C \min(|S||\Sigma|, |S|^{1/d}w(\Sigma), w(S)|\Sigma|^{1/d})} (\int_{\R^d\setminus S}|f(x)|^2 dx + \int_{\R^d\setminus\Sigma}|\hat{f}(x)|^2 dx) $$ where $\hat{f}$ is the Fourier transform of $f$ and $w(\Sigma)$ is the mean width of $\Sigma$. This extends to dimension $d\geq 1$ a result of Nazarov \cite{pp.Na} in dimension $d=1$.

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