Stieltjes Integral Theorem and The Hilbert Transform

TL;DR

This paper explores the application of the Stieltjes integral theorem to the Hilbert transform, resulting in an alternative, computationally faster integral definition and insights into Fourier domain properties, with extensions into complex analysis.

Contribution

It introduces a novel integral formulation of the Hilbert transform using Stieltjes theorem, enhancing computational efficiency and providing a new perspective through complex analysis.

Findings

Derived an alternative integral form of the Hilbert transform.

Demonstrated accelerated computation via Fourier domain analysis.

Extended the integral approach into complex analysis with a logarithmic kernel.

Abstract

Stieltjes integral theorem is more commonly known by the phrase 'integration by parts' and enables rearrangement of an otherwise intractable integral to a more amenable form; often permitting completion of an integral in closed form. Applying Stieltjes integral theorem to the Hilbert transform yields an equivalent alternate integral definition, which is homeomorphic and exhibits accelerated computation. By virtue of the convolution theorem, the integral is mapped to Fourier image space and delineates requirements for the inverse Fourier transform, also, these requirements reveal the underlying reason for increased computational speed. Lastly, an alternative to Cauchy's integral formula is deduced by extending the line integral into the complex domain and involves a line integral with logarithmic kernel.