A caveat about applications of the unilateral Fourier transform

Antonio S De Castro

doi:10.1590/1806-9126-rbef-2020-0129

Abstract

Abstract It is presented a warning about the erroneous use of unilateral Fourier transform with nonhomogeneous Dirichlet or Neumann boundary conditions in a well-known textbook on integral transforms, and also in a few papers recently diffused in the literature.

Highlights

Integral transforms are used in a variety of applications, for example, to evaluate certain definite integrals, to transform a partial differential equation into an ordinary differential equation, to transform an ordinary differential equation into a simpler differential equation or into an algebraic equation, and they can play a more theoretical role in applied problems
Bhatta wrote in the Preface to the Second Edition of their book [1]: “When the first edition of this book was published in 1995 under the sole authorship of Lokenath Debnath, it was well received, and has been used as a senior undergraduate or graduate level text and research reference in the United States and abroad for the last ten years.”. It is really a wellknown book, in its third edition [2]. It is a book marked by great carelessness concerning the unilateral Fourier transform and its applications which has apparently triggered the emergence of a number of papers [3]- [11] emulating the misuses found there
The use of integral transforms is worthless if their inversion formulas fail

Summary

Introduction

Integral transforms are used in a variety of applications, for example, to evaluate certain definite integrals, to transform a partial differential equation into an ordinary differential equation, to transform an ordinary differential equation into a simpler differential equation or into an algebraic equation, and they can play a more theoretical role in applied problems. Bhatta wrote in the Preface to the Second Edition of their book [1]: “When the first edition of this book was published in 1995 under the sole authorship of Lokenath Debnath, it was well received, and has been used as a senior undergraduate or graduate level text and research reference in the United States and abroad for the last ten years.” It is really a wellknown book, in its third edition [2]. The Fourier sine and cosine transforms of f (x) are denoted by Fs {f (x)} = Fs (k) and Fc {f (x)} = Fc (k), respectively, and are defined by the integrals

Results

Conclusion