## Harmonic Analysis: A Gentle IntroductionMany branches of mathematics come together in harmonic analysis, each adding richness to the subject and each giving insights into this fascinating field. Devito's Harmonic Analysis presents a comprehensive introduction to Fourier analysis and Harmonic analysis and provides numerous examples and models so that students leave with a clear understanding of the theory. |

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### Contents

Preliminaries | 1 |

Relations and Functions | 4 |

The Real Number System | 7 |

The Complex Number System | 11 |

Analysis | 15 |

Classical Harmonic Analysis | 23 |

The Dirichlet Problem for a Disk | 24 |

Continuous Functions on the Unit Circle | 25 |

Infinite Orthonormal Sets Hilbert Space | 96 |

The Completion | 103 |

Wavelets | 107 |

The Fourier Transform | 113 |

Invertible Elements in f1 Z | 119 |

The Fourier Transform on R | 123 |

Naive Group Theory | 128 |

Not So Naive Group Theory | 131 |

The Method of Fourier | 29 |

Uniform Convergence | 32 |

The Formulas of Euler | 39 |

Cesaro Convergence | 43 |

Fejers Theorem | 47 |

At Last the Solution | 52 |

Extensions of the Classical Theory | 57 |

Functions on Other Intervals | 61 |

Functions with Special Properties | 65 |

Pointwise Convergence of the Fourier Series | 70 |

Fourier Series in Hilbert Space | 79 |

Normed Vector Spaces | 80 |

Convergence in Normed Spaces | 84 |

Inner Product Spaces | 90 |

### Common terms and phrases

abelian group additive group algebra called Cauchy sequence Cesaro means Chapter Zero choose Clearly closed set complex numbers Compute contains continuous function converges uniformly Corollary Cr(T Definition denote dense discussion element equation equivalence classes Euler Exercises finite fixed fn(x follows Fourier series Fourier transform function ip functions defined given harmonic in G hence Hilbert space inner product space interval ip(a ip(g Lemma Let G linear map linear subspace M(ip multiplicative group multiplicative inverse natural number nonempty set nonempty subset normed space Note one-to-one open set orthonormal basis orthonormal set polynomial positive integer prime problem Proof proper ideal properties prove real-valued function Recall root of unity scalar Section series converges smallest positive solutions subgroup Suppose Theorem trigonometric uniformly continuous unit circle vector space wavelet write zero vector