A functional model for the Fourier–Plancherel operator truncated to the positive semiaxis

Abstract:

The truncated Fourier operator $\mathscr {F}_{\mathbb {R}^+}$, \begin{equation*}(\mathscr {F}_{\mathbb {R}^+}x)(t)=\frac {1}{\sqrt {2\pi }}\int _{\mathbb {R}^+}x(\xi )e^{it\xi } d\xi ,\quad t\in {\mathbb {R}^+}, \end{equation*} is studied. The operator $\mathscr {F}_{\mathbb {R}^+}$ is viewed as an operator acting in the space $L^2(\mathbb {R}^+)$. A functional model for the operator $\mathscr {F}_{\mathbb {R}^+}$ is constructed. This functional model is the operator of multiplication by an appropriate (${2\times 2}$)-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.