Introduction

In this chapter we make some general remarks about Fourier transformation before proceeding to a systematic statement of theorems in the remaining chapters. The aim will be to progress from simple situations towards more complicated ones, and we start in chapter 7 with the socalled good functions. We then progressively consider wider classes of functions and at each stage show in what way the theorems need to be modified. Throughout chapters 7–11 we retain the restriction that both the function and its Fourier transform are locally integrable functions defined almost everywhere on the real line; such pairs we will call ordinary Fourier pairs. Throughout we will assume the use of Lebesgue integration (see chapter 2) unless otherwise stated, and the functions in an ordinary Fourier pair can display fairly ‘pathological’ local behaviour. However, it is possible to consider ‘even worse’ situations, and in chapters 12-14 we show how Fourier transformation can be applied to entities which are not locally integrable and are not even defined through their values almost everywhere on the real line. These entities are the generalized functions, and we will introduce several examples of these including the well known Dirac delta function.

Although in stating theorems it is necessary to distinguish clearly between those which apply to generalized functions and those which apply to ordinary functions, we wish to stress the view that the introduction of generalized functions is only one step in a sequence of generalizations, and that the ordinary pairs can be regarded as special cases within the generalized theory.

Introduction

The Dirac delta function δ(x) is the best known of a class of entities called generalized functions. The generalized functions are important in Fourier theory because they allow any function in LLOC (and indeed any generalized function also) to be Fourier transformed. Thus the function f(x) = 1 has no Fourier transform within the realm of functions in LLOC, but it acquires the transform δ(y) in the generalized theory. The generalized functions thus remove a blockage which existed in the previous theory. There is an analogy with the way in which the use of complex numbers allows any quadratic equation to be solved, whilst within the realm only of real numbers not all quadratic equations have solutions.

Generalized functions remove many other blockings which occur in the analysis of functions in LLOC. For instance, every locally integrable function (and indeed every generalized function) can be regarded as the integral of some generalized function and thus becomes infinitely differentiate in this new sense. Many sequences of functions which do not converge in any accepted sense to a limit function in LLOC can be regarded as converging to a generalized function, and moreover in this case the sequence of Fourier transforms will necessarily converge to the Fourier transform of the limit. Thus, in many ways the use of generalized functions simplifies the rules of analysis.

Introduction

Early developments of Fourier theory were based on the theory of integration of G. F. B. Riemann (1826–66). However, an alternative approach to integration developed by H.L. Lebesgue (1875–1941) turns out to be more powerful and simpler to use so far as Fourier theory is concerned, and virtually all modern mathematical approaches are based on Lebesgue's theory. Whilst a full understanding of Lebesgue's theory is not necessary in later chapters, some familiarity with the ideas is essential, and the following description is intended for a reader who is familiar with Riemann but not Lebesgue integration.

Riemann integration

The procedure for defining the proper Riemann integral of some single valued, real function of a real variable between the finite limits a and b may be visualized by dividing the area under the graph of the function into vertical strips. The area of each strip is approximated to the product of the width of the strip and the sampled value of the function at some arbitrary point within the strip, and one considers the limit of the sum of these approximate areas as the widths of the strips tend to zero and the number of such strips tends to infinity.

A rigorous treatment, as in Apostol (1974) for instance, considers whether the limit so obtained depends upon the particular way the strips or the sample values are chosen at each stage of the limiting process.

This handbook is intended to assist those scientists, engineers and applied mathematicians who are already familiar with Fourier theory and its applications in a non-rigorous way, but who wish to find out the exact mathematical conditions under which particular results can be used. A reader is assumed whose mathematical grounding in other respects goes no further than the traditional first year university course in mathematics taken by physical scientists or engineers. Advances in mathematical sophistication have led to a growing divide between those books intended for mathematics specialists and those intended for others, and this handbook represents a conscious effort to bridge this gap.

The core of the book consists of rigorous statements of the most important theorems in Fourier theory, together with explanatory comments and examples, and this occupies chapters 6–16. This is preceded, in chapters 1–5, by an introduction to the terminology and the necessary ideas in mathematical analysis including, for instance, the interpretation of Lebesgue integrals. Proofs of theorems are not provided, and the first part is not intended as a complete grounding in mathematical analysis; however, it is intended that the book should be self contained and that it should provide a background which will assist the interested reader to follow proofs in standard mathematical texts.

In the chapters on Fourier theorems, those classical theorems dealing with locally integrable functions are covered first in a way that leads naturally to the later sections covering generalized functions.

Introduction

It would be apparent from chapters 3 and 4 that analytic methods for nonlinear problems of diffusion (in common with all nonlinear problems) have severe limitations. For example, the non-planar N wave solution obtained by Crighton and Scott (1979) via matched asymptotic expansions, undoubtedly useful in the Taylor-shock regime, holds only over a certain finite time as detailed in chapter 3. The evolutionary shock regime after the Taylor shock and the subsequent decay later to the linear form are scarcely covered by analysis. The self-similar solutions discussed in chapter 4, although the only genuine exact solutions of nonlinear problems (when they exist), are special in nature. They may not exist for a given problem. Even when a self-similar form exists, it satisfies only special initial/boundary conditions so that a given physical problem dictating specific initial and boundary conditions will, in general, not have a self-similar solution. The latter almost always satisfies some singular initial conditions signifying its asymptotic nature. One naturally turns, therefore, to numerical techniques to get a clear qualitative as well as quantitative picture of the phenomena over the entire course of the wave. Since the problems that we discuss involve shocks with discontinuous or steep-fronted initial conditions and have an infinitely long time domain, the numerical methods must be sturdy enough to meet these exigencies. The finite difference or pseudo-spectral methods that may be used for such problems should have other necessary attributes, namely stability, convergence, small truncation error, in addition to economy in computational time in view of the large evolution time of the wave.

I was fortunate to receive help from many. Professor Sir James Lighthill FRS provided the impetus to undertake this venture. Dr Allen Tayler helped my rather amorphous ideas assume a precise form. Professor A. Richard Seebass, many years ago, introduced me to the mysteries of Burgers' equation and its kindred class. I had the benefit of very fruitful discussions with Professors D. G. Crighton and J. D. Murray. Professor Colin Rogers read through parts of the manuscript. Professors P. N. Kaloni and M. C. Singh were excellent hosts during my sojourns at the Universities of Windsor and Calgary, where I wrote part of the book. My student, Mr K. R. C. Nair, carried out much of the computations reported in the final chapter and assisted in various other ways. Ms Thelma Stanley, with good cheer and patience, typed the manuscript and many a change I made in it. Finally, I owe a lot to my wife Rita, who cared and comforted.

I gratefully acknowledge the financial assistance provided for the preparation of the manuscript by the Curriculum Development Cell, established at the Indian Institute of Science by the Ministry of Education and Culture, Government of India.

In conclusion, I thank Dr David Tranah of Cambridge University Press, for his courtesy and thoughtful consideration during the entire course of the publication of this monograph.