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

TL;DR

This paper develops a functional model for the truncated Fourier operator on the positive half-axis, enabling the analysis of its spectrum and resolvent in the space of square-integrable functions.

Contribution

A novel functional model for the truncated Fourier operator is constructed, facilitating spectral analysis and resolvent estimation.

Findings

Spectrum of the truncated Fourier operator is explicitly determined.

The resolvent is estimated near the spectrum.

The model simplifies analysis of the operator's properties.

Abstract

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.