Norming Sets

Abstract

The main objective of this report is the study of the Logvinenko-Sereda sets for different function spaces. It consists in characterizing the subsets $G \subset \Omega$ such that there is a constant $C>$ 0 where $\|f\|^2\leq C\int_{g}|f|^{2} dm$. Following to the proof that appears in the book of V. Havin and B. Jöricke we have obtained the Logvinenko-Sereda theorem for the Paley-Wiener space. Moreover, for the same function space we have found another argument based on the proof of Daniel H. Luecking for the Bergman space in the ball $B=\{x\in\mathbb{R}^{n}:|x|<1\}$. In this case, we have taken the same structure of the proof with the translations group and euclidean balls instead of the automorphism group and hyperbolic balls. Next, considering the same idea as for the Paley-Wiener space we have achieved the Logvinenko-Sereda theorem for the Classic Fock space. Finally, we have finished with the analogous result for the space of polynomials in the torus.