Analogues of the Laplace Transform and Z-Transform with Piecewise Linear Kernels

Marianito R Rodrigo,Mandy Li

doi:10.3390/foundations1010008

Abstract

Two new transforms with piecewise linear kernels are introduced. These transforms are analogues of the classical Laplace transform and Z-transform. Properties of these transforms are investigated and applications to ordinary differential equations and integral equations are provided. This article is ideal for study as a foundational project in an undergraduate course in differential and/or integral equations.

Highlights

An integral transform maps a function from its original function space into another function space via integration
Properties of the original function might be more characterised in the transformed space rather than in the original space
Integral transforms arise in many areas of mathematics, e.g., differential and integral equations, probability, number theory and computer science

Summary

Introduction

An integral transform maps a function from its original function space into another function space via integration. The kernel Ks (t) = e−st of the Laplace transform is positive and decreasing for s, t ∈ R+ , and satisfies. The kernel Kz ( j) = z− j of the Z-transform is positive and decreasing for z ∈ R+. Suppose that we replace the above two kernels by behaved but piecewise linear functions. The goal of this article is to investigate some properties and applications of these analogues of the Laplace transform and Z-transform with piecewise linear kernels. The results in this article are useful as a springboard for further investigation of other continuous and discrete transforms, and are useful as a foundational project for a first course in differential and/or integral equations

Properties of the Continuous Transform

Examples

Properties of the Discrete Transform

Discussion