A Generalised Continuous Primitive Integral and Some of Its Applications

TL;DR

This paper introduces a new Laplace-based integral that generalizes the Perron integral, explores its properties, and demonstrates its applications in differential equations and harmonic analysis.

Contribution

It defines a Laplace integral that extends the Perron integral, proves key properties, and applies it to differential equations and Poisson integrals.

Findings

Laplace integral includes Perron integral but is strictly more general.

The space of Laplace integrable functions is incomplete in Alexiewicz's norm.

Applications to differential equations and Poisson integral are demonstrated.

Abstract

Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hake's theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylor's theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewicz's norm, the space of all Laplace integrable functions is incomplete and contains the set of all polynomials densely. Applications are shown to Poisson integral, a system of generalised ordinary differential equations and higher-order generalised ordinary differential equation.