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