## Fourier Transforms |

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### Contents

The Fourier transform on L1 | 6 |

The Fourier transform on L2 | 43 |

Generalizations of Wieners theorem | 52 |

3 other sections not shown

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### Common terms and phrases

absolutely convergent addition analytic arbitrary assume bounded chapter character choose closed ideal closed maximal ideals closure coincides compact completes the proof complex conclude contains continuous on 00,00 convolution COROLLARY define the Fourier defined DEFINITION denoted e-iux eixt element equal equivalent establish example exists f e L1 fact fell Finally follows Fourier transform function function f given Hence holds identically Iffe immediately implies important integral interval inversion LCAG Lebesgue lemma limit point linear mean measure mentioned Moreover multiplication neighborhood non-decreasing norm Note obtain operation particular positive integer preceding PROOF prove reader real numbers reference REMARK requirements respect result right side sequence shown SR(u statement sufficient summable suppose THEOREM topology translates true vanishes verified yields zero