A functional model for the Fourier--Plancherel operator truncated on the positive half-axis

Abstract

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

17 pages. arXiv admin note: text overlap with arXiv:1208.3817