Suppose \(n\in \mathbb{N}\), \(\beta_1,\ldots ,\beta_n\in \mathbb{C}\) and \(\lambda_1,\ldots ,\lambda_n\in \mathbb{C}\). Let also \(p(t)=\sum_{k=1}^{n} \beta_k \e^{\lambda_k t}\) be an exponential polynomial, \(I\subseteq \mathbb{R}\) be a bounded interval and \(S\subseteq I\) a measurable subset with positive measure. Then there is a constant \(C>0\) such that
\[ \sup_{t\in I} \lvert p(t)\rvert \le \e^{\lvert I\rvert\max_{k=1,\ldots ,n} \lvert \Re \lambda_k\rvert}\biggl(\frac{C\lvert I\rvert}{\lvert S\rvert}\biggr)^{n-1}\sup_{t\in S} \lvert p(t)\rvert \][1, Theorem 1].
See also Link to heading
References Link to heading
- F. Nazarov,
Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type,
Rossiĭskaya Akademiya Nauk. Algebra i Analiz, vol. 5, no. 4, p. 3–66, 1993.