Modified Stieltjes Transform and Generalized Convolutions

TL;DR

This paper introduces a modified Stieltjes Transform that generalizes classical transforms, enabling the development of new classes of both commutative and non-commutative generalized convolutions.

Contribution

The paper proposes a novel modification of the Stieltjes Transform to unify and extend the concepts of Stieltjes and Fourier transforms, leading to new convolution operations.

Findings

New classes of generalized convolutions introduced

Unified framework for Stieltjes and Fourier transforms

Potential applications in probability and harmonic analysis

Abstract

Classical Stieltjes Transform is modified in a way to generalize both Stieltjes and Fourier transforms. This transform allows to intro- duce new classes of commutative and non-commutative generalized convolutions. Key words: Stieltjes Transform; characteristic function; generalized convolution.