## An Introduction to Fourier Analysis and Generalised FunctionsThis 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. |

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### Contents

Introduction | 1 |

12 Knowledge assumed of the reader | 2 |

introductory remarks | 3 |

introductory remarks | 8 |

introductory remarks | 10 |

The theory of generalised functions and their Fourier transforms | 15 |

22 Generalised functions The delta function and its derivatives | 16 |

23 Ordinary functions as generalised functions | 21 |

33 Integral powers | 35 |

34 Integral powers multiplied by logarithms | 40 |

35 Summary of Fourier transform results | 42 |

The asymptotic estimation of Fourier transforms | 46 |

42 Generalisations of the RiemannLebesgue lemma | 47 |

43 The asymptotic expression for the Fourier transform of a function with a finite number of singularities | 51 |

Fourier series | 58 |

52 Determination of the coefficients in a trigonometrical series | 60 |

24 Equality of a generalised function and an ordinary function in an interval | 25 |

25 Even and odd generalised functions | 26 |

26 Limits of generalised functions | 27 |

Definitions properties and Fourier transforms of particular generalised functions | 30 |

32 Nonintegral powers multiplied by logarithms | 34 |

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### Common terms and phrases

absolutely convergent absolutely integrable applied asymptotic behaviour asymptotic expansion asymptotic expression behaved at infinity boundary conditions Cauchy principal value chapter completes the proof condition of definition continuous function cosine series defined definition 14 delta function differentiable any number equation 12 error term EXERCISE exists F.T. of f(x fairly good function Find an asymptotic finite interval finite limit finite number Fm(x Fourier cosine Fourier integrals Fourier series Fourier sine series Fourier's inversion theorem generalised function f(x given function Gm(y Hence Iff(x infinite Integral powers LAURENT SCHWARTZ linear combination Non-integral powers Note number of singularities obtained odd function oo to oo period 2l periodic function periodic generalised function precise order PROOF OF CONSISTENCY reader regular sequence Riemann-Lebesgue lemma sequence fn(x sgnx solution taking Fourier transforms tends to zero term by term test functions theorem 15 theorem 20 theory of Fourier theory of generalised trigonometrical series vanishes whence