## Introduction to Fourier Analysis on Euclidean SpacesThe authors present a unified treatment of basic topics that arise in Fourier analysis. Their intention is to illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations, and to motivate the study of harmonic analysis on more general spaces having an analogous structure, e.g., symmetric spaces. |

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### Contents

The Fourier Transform | 1 |

Boundary Values of Harmonic Functions | 37 |

The Theory of H Spaces on Tubes | 89 |

Symmetry Properties of the Fourier Transform | 133 |

Singular Integrals and Systems of Conjugate | 217 |

Multiple Fourier Series | 245 |

287 | |

295 | |

### Common terms and phrases

analytic function apply argument Banach space belongs Bochner Borel measure boundary values bounded Chapter characteristic function cone consider continuous function convergence convexity theorem convolution Corollary denote domain equality established everywhere example exists a constant extend fact finite Borel measure following result follows immediately Fourier series Fourier transform function defined function F harmonic functions Hilbert transform immediate consequence implies inequality interpolation theorem L2-norm Lemma linear operators Ll(En mapping Marcinkiewicz theorem maximal function Moreover multiplier nonnegative nontangential limits norm obtain orthogonal Plancherel theorem Poisson integral Poisson kernel Poisson summation formula polynomials proof of Theorem proved real numbers restricted Riesz transforms rotation satisfying sequence singular integral operators spherical harmonics subset suffices to show summability tempered distribution tends testing function Theorem 1.3 theory tube unit sphere variables weak type Zygmund