Geodesic
The truncated Fourier operator. General results
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 8 (2012) no. 2, pp. 158-176 
V. Katsnelson; R. Machluf. The truncated Fourier operator. General results. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 8 (2012) no. 2, pp. 158-176. http://geodesic.mathdoc.fr/item/JMAG_2012_8_2_a3/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\mathcal F$ be the one dimensional Fourier–Plancherel operator and $E$ be a subset of the real axis. The truncated Fourier operator is the operator $\mathcal F_E$ of the form $\mathcal F_E=P_E\mathcal FP_E$, where $(P_Ex)(t)=\mathbf 1_E(t)x(t)$, and $\mathbf 1_E(t)$ is the indicator function of the set $E$. In the presented work, the basic properties of the operator $\mathcal F_E$ according to the set $E$ are discussed. Among these properties there are the following ones: 1) the operator $\mathcal F_E$ has a nontrivial null-space; 2) $\mathcal F_E$ is strictly contractive; 3) $\mathcal F_E$ is a normal operator; 4) $\mathcal F_E$ is a Hilbert–Schmidt operator; 5) $\mathcal F_E$ is a trace class operator.

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