Let \(J\) be an interval with \(|J|=b\). If \(f\in L^p(\mathbb{R})\), \(p \in [1,\infty)\), and \(\supp \hat{f} \subset J\), then on every good interval \(I\) of \(f\) with length \(a\) a point \(x\in I\) exists, such that for every \(y\in D(x,R)=\{y\in \mathbb{C}\mid |x-y|< R\}\) with \(R>0\) we obtain
\begin{equation}\label{eq:sup_est} |f(y)|\le \Bigl(\frac{2}{a}\Bigr)^{1/p}\exp(CbR)\lVert f\rVert_{L^p(I)}, \end{equation}where \(C>0\) is some numerical constant [1].
Proof
Every good intervals has a good point
\(x\). We use this point as the expansion point of the Taylor expansion of \(f\).
By estimating the derivatives with the good point property, we obtain an exponential series and finally \eqref{eq:sup_est}.
Remarks
- This Lemma especially implies \begin{equation*} \sup_{y\in D(x,R)} |f(y)|\le 2^{\frac{1}{p}}\exp(CabR)\lVert f\rVert_{L^p(I)}, \end{equation*}
- This Lemma is needed to prove Kovrijkine’s uncertainty principle in one dimension .
See also Link to heading
References Link to heading
- O. Kovrijkine,
Some results related to the Logvinenko-Sereda theorem,
Proceedings of the American Mathematical Society, vol. 129, no. 10, pp. 3037–3047, 2001. doi:10.1090/S0002-9939-01-05926-3