Classical Fourier Transforms

§This book gives a thorough introduction on classical Fourier transforms in a compact and self-contained form. Chapter I is devoted to the L1-theory: basic properties are proved as well as the Poisson summation formula, the central limit theorem and Wiener's general tauberian theorem. As an illustration of a Fourier transformation of a function not belonging to L1-theory an integral due to Ramanujan is given. Chapter II is devoted to the L2-theory, including Plancherel's theorem, Heisenberg's inequality, the Paley-Wiener theorem, Hardy's interpolation formula and two inequalities due to Bernstein. Chapter III deals with Fourier-Stieltjes transforms. After the basic properties are explained, distribution functions, positive-definite functions and the uniqueness theorem of Offord are treated. The book is intended for undergraduate students and requires of them basic knowledge in real and complex analysis. 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 Poisson's summation formula . . . . . . . . . . 32 §6. The uniqueness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .