Classical Fourier Transforms

In 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 Poissons summation formula . . . . . . . --. --. . 32 6. The uniqueness theorem . . . . . . . -. . . . . . . . . . . -. . . . -. . . . . . . . . . . .