## An Introduction to Harmonic AnalysisFirst published in 1968, An Introduction to Harmonic Analysis has firmly established itself as a classic text and a favorite for students and experts alike. Professor Katznelson starts the book with an exposition of classical Fourier series. The aim is to demonstrate the central ideas of harmonic analysis in a concrete setting, and to provide a stock of examples to foster a clear understanding of the theory. Once these ideas are established, the author goes on to show that the scope of harmonic analysis extends far beyond the setting of the circle group, and he opens the door to other contexts by considering Fourier transforms on the real line as well as a brief look at Fourier analysis on locally compact abelian groups. This new edition has been revised by the author, to include several new sections and a new appendix. |

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### Contents

Fourier Series on T page | 1 |

The Convergence of Fourier Series | 67 |

The Conjugate Function | 83 |

Interpolation of Linear Operators | 117 |

Lacunary Series and Quasianalytic Classes | 133 |

Fourier Transforms on the Line | 151 |

Fourier Analysis on Locally Compact Abelian Groups | 223 |

Commutative Banach Algebras | 231 |

VectorValued Functions | 295 |

B Probabilistic Methods | 299 |

307 | |

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### Common terms and phrases

abelian group absolutely continuous almost-periodic analytic AP(R arbitrary assume bounded function Chapter clearly compact support complex numbers condition consequently contains continuous functions convolution Corollary defined denote dense dual equivalent EXERCISES FOR SECTION exponential Fejer kernel Fejer's Fourier coefficients Fourier series Fourier transform Fourier-Stieltjes coefficients Fourier-Stieltjes transform func function F Haar measure Hausdorff space hence Hint holomorphic homeomorphism homogeneous Banach space implies inequality infinitely differentiable interval lacunary LCA group Lebesgue Lemma Let f e Let G linear functional Ll(T LP(R LP(T LX(R LX(T maximal ideal maximal ideal space multiplicative linear functional neighborhood nonnegative obtain open set orthogonal Parseval's formula Plancherel's theorem pointwise Poisson integral positive definite positive measure prove real numbers real-valued Remark satisfying sequence set of divergence Show spectral synthesis subset subspace summability kernel trigonometric polynomials trigonometric series uniformly continuous valid valued vanishes weak-star topology write