## Distributions, complex variables, and Fourier transforms |

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### Contents

lntroduction | 1 |

Schwartz distributions | 11 |

Regularization localization and support of distributions | 20 |

Copyright | |

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### Common terms and phrases

Analogously analytic function analytic representation antiderivative boundary values bounded Cauchy kernel Cauchy representation Chapter characteristic function Cm)-function compact set compact support complex valued complex variables contained continuous function converges to zero converges uniformly convolution Corollary defined Definition denote dispersion relation distribution in 2D distribution with compact example finite order derivative Fubini's theorem func function f function in 2D harmonic function Heaviside function Hence lim identity immediate consequence implies inverse Fourier transform L2-functions Let f Let f(t light cone linear functional multiplier n-harmonic Note obtain open set Parseval's formula partition of unity Plancherel's theorem polynomial Problem Proof proves the lemma proves the theorem quadrupole real axis Representation of distributions Section sense of 2D tempered distribution tempered function tends to zero tion topology uniformly continuous upper half-plane valued functions vanishes z-plane