  Let (\mathscr{F}) be the one dimensional Fourier-Plancherel operator and (E)
be a subset of the real axis. The truncated Fourier operator is the operator
(\mathscr{F}_E) of the form (\mathscr{F}_E=P_E\mathscr{F}P_E), where
((P_Ex)(t)=\chi_E(t)x(t)), and (\chi_E(t)) is the indicator function of the set
(E). In the presented first part of the work, the basic properties of the
operator (\mathscr{F}_E) according to the set (E) are discussed.
  Among these properties there are the following one. The operator
(\mathscr{F}_E): 1. has a not-trivial null-space; 2. is strictly contractive;
3. is a normal operator; 4. is a Hilbert-Schmidt operator; 5. is a trace class
operator.
