The truncated Fourier operator. I

TL;DR

This paper investigates the fundamental properties of the truncated Fourier operator, including its null-space, contractiveness, normality, and classification as Hilbert-Schmidt and trace class, based on the subset of the real axis used for truncation.

Contribution

It provides a detailed analysis of the basic properties of the truncated Fourier operator depending on the set E, including its null-space, contractiveness, normality, and operator class.

Findings

The operator has a non-trivial null-space.

The operator is strictly contractive.

The operator is normal, Hilbert-Schmidt, and trace class.

Abstract

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.