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7932 results in Fluid dynamics and solid mechanics

Page 397 of 397

The ANZIAM Journal

  • 1 - Introduction

  • M. J. Lighthill
  • Book:
    An Introduction to Fourier Analysis and Generalised Functions
    Published online:
    05 March 2012
    Print publication:
    01 January 1958, pp 1-14

  • Summary

    Scope and purpose of the book

    This book grew out of an undergraduate course in the University of Manchester, in which the author attempted to expound the most useful facts about Fourier series and integrals. It seemed to him on planning the course that a satisfactory account must make use of functions like the delta function of Dirac which are outside the usual scope of function theory. Now, Laurent Schwartz in his Théorie des distributions* has evolved a rigorous theory of these, while Professor Temple has given a version of the theory (Generalised functions) which appears to be more readily intelligible to students. With some slight further simplifications the author found that the theory of generalised functions was accessible to undergraduates in their final year, and that it greatly curtails the labour of understanding Fourier transforms, as well as making available a technique for their asymptotic estimation which seems superior to previous techniques. This is an approach in which the theory of Fourier series appears as a special case, the Fourier transform of a periodic function being a ‘row of delta functions’.

    The book which grew out of the course therefore covers not only the principal results concerning Fourier transforms and Fourier series, but also serves as an introduction to the theory of generalised functions, whose general properties as well as those useful in Fourier analysis are derived, simply but without any departure from rigorous standards of mathematical proof.

An Introduction to Fourier Analysis and Generalised Functions
  • M. J. Lighthill
  • Published online:
    05 March 2012
    Print publication:
    01 January 1958
  • This monograph on generalised functions, Fourier integrals and Fourier series is intended for readers who, while accepting that a theory where each point is proved is better than one based on conjecture, nevertheless seek a treatment as elementary and free from complications as possible. Little detailed knowledge of particular mathematical techniques is required; the book is suitable for advanced university students, and can be used as the basis of a short undergraduate lecture course. A valuable and original feature of the book is the use of generalised-function theory to derive a simple, widely applicable method of obtaining asymptotic expressions for Fourier transforms and Fourier coefficients.

Page 397 of 397