  The truncated Fourier operator $\mathscr{F}_{\mathbb{R^{+}}}$, $$
(\mathscr{F}_{\mathbb{R^{+}}}x)(t)=\frac{1}{\sqrt{2\pi}}
\int\limits_{\mathbb{R^{+}}}x(\xi)e^{it\xi}\,d\xi\,,\ \ \
t\in{}{\mathbb{R^{+}}}, $$ is studied. The operator
$\mathscr{F}_{\mathbb{R^{+}}}$ is considered as an operator acting in the space
$L^2(\mathbb{R^{+}})$. The functional model for the operator
$\mathscr{F}_{\mathbb{R^{+}}}$ is constructed. This functional model is the
multiplication operator on the appropriate $2\times2$ matrix function acting in
the space $L^2(\mathbb{R^{+}})\oplus{}L^2(\mathbb{R^{+}})$. Using this
functional model, the spectrum of the operator $\mathscr{F}_{\mathbb{R^{+}}}$
is found. The resolvent of the operator $\mathscr{F}_{\mathbb{R^{+}}}$ is
estimated near its spectrum.
