One-parameter groups of operators and discrete Hilbert transforms

TL;DR

This paper demonstrates that the discrete Hilbert transform and the discrete Kak-Hilbert transform serve as infinitesimal generators for one-parameter groups of operators, linking these transforms to continuous symmetry groups.

Contribution

It establishes a novel connection between discrete Hilbert transforms and the theory of one-parameter operator groups, providing new insights into their structural properties.

Findings

Discrete Hilbert transform is an infinitesimal generator of a one-parameter group.

Discrete Kak-Hilbert transform also generates a one-parameter group.

The work bridges discrete transforms with continuous operator group theory.

Abstract

We show that the discrete Hilbert transform and the discrete Kak-Hilbert transform are infinitesimal generator of one-parameter groups of operators