## Classical Fourier TransformsIn gratefuZ remerribrance of Marston Morse and John von Neumann This text formed the basis of an optional course of lectures I gave in German at the Swiss Federal Institute of Technology (ETH), Zlirich, during the Wintersemester of 1986-87, to undergraduates whose interests were rather mixed, and who were supposed, in general, to be acquainted with only the rudiments of real and complex analysis. The choice of material and the treatment were linked to that supposition. The idea of publishing this originated with Dr. Joachim Heinze of Springer Verlag. I have, in response, checked the text once more, and added some notes and references. My warm thanks go to Professor Raghavan Narasimhan and to Dr. Albert Stadler, for their helpful and careful scrutiny of the manuscript, which resulted in the removal of some obscurities, and to Springer-Verlag for their courtesy and cooperation. I have to thank Dr. Stadler also for his assistance with the diagrams and with the proof-reading. Zlirich, September, 1987 K. C. Contents Chapter I. Fourier transforms on L (-oo,oo) 1 §1. Basic properties and examples •. •••••. . ••. . •. •. . . •. •. . •. . . . • 1 §2. The L 1-algebra ••. . . . . . . ••••. . ••. •. . ••. . ••. . •. . . ••. . . . ••. •. . 16 §3. Differentiabili ty properties . . . •••. •. •••••••. . . . ••••. •. . . •. 18 §4. Localization, Mellin transforms . . . . . . •. •. . . . . . •. . . . . . •. . •. . 25 §5. Fourier series and Poisson's summation formula . . . . . . . ••. ••. . 32 §6. The uniqueness theorem . . . . . . . •. . . . . . . . . . . •. . . . •. . . . . . . . . . . . |

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### Contents

Fourier transforms on In 0000 | 1 |

2 The L1algebra | 2 |

3 Differentiability properties | 18 |

Copyright | |

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apply assume assumption Banach space belongs Bochner bounded bounded variation CALIFORNIA Cauchy principal value Ch.I Chapter choose closed coefficient complex condition consider constant converges convolution Corollary cosh defined definition denotes derivative DIEGO DIEGO differentiable differs equals ERSITY everywhere Example exists a function f is continuous f(x+t finite fixed follows formula Fourier series Fourier transform function f further given gives Hence holds hypothesis implies inequality infinitely instance integral interval Lemma LIBRARY linear measurable non-decreasing norm obtain origin particular points of continuity positive Proof prove referred relation Remarks result SAN DIEGO satisfies sequence side similarly space sufficient summability Theorem Theorem 11 theory uniformly uniqueness UNIVERSITY vanishes variable Wiener zero