Nazarov's uncertainty principles in higher dimension

Philippe Jaming (1)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

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 $f\in L^2(\R^d)$, $$ \int_{\R^d}|f(x)|^2\,\mbox{d}x\leq C e^{C\min\bigl(|S||\Sigma|,|S|^{1/d}w(\Sigma),w(S)|\Sigma|^{1/d}\bigr)}\left( \int_{\R^d\setminus S}|f(x)|^2\,\mbox{d}x+\int_{\R^d\setminus\Sigma}|\widehat{f}(x)|^2\,\mbox{d}x\right) $$ where $\widehat{f}$ 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 $d=1$.

Mots clés: strong annihilating pair; random periodization; uncertainty principle

Domaines: Analyse classique [math.CA]

Philippe Jaming. Nazarov's uncertainty principles in higher dimension. Journal of Approximation Theory, 2007, pp.doi:10.1016/j.jat.2007.04.005. ⟨10.1016/j.jat.2007.04.005⟩. ⟨hal-00120268⟩