Classical Fourier Transforms

ISBN-13: 9783540502487 / Angielski / Miękka / 1988 / 172 str.

Komaravolu Chandrasekharan

Opis

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ."

Spis treści

I. Fourier transforms on L1 (-?,?).- §1. Basic properties and examples.- §2. The L1 -algebra.- §3. Differentiability properties.- §4. Localization, Mellin transforms.- §5. Fourier series and Poisson’s summation formula.- §6. The uniqueness theorem.- §7. Pointwise summability.- §8. The inversion formula.- §9. Summability in the L1-norm.- §10. The central limit theorem.- §11. Analytic functions of Fourier transforms.- §12. The closure of translations.- §13. A general tauberian theorem.- §14. Two differential equations.- §15. Several variables.- II. Fourier transforms on L2(-?,?).- §1. Introduction.- §2. Plancherel’s theorem.- §3. Convergence and summability.- §4. The closure of translations.- §5. Heisenberg’s inequality.- §6. Hardy’s theorem.- §7. The theorem of Paley and Wiener.- §8. Fourier series in L2(a,b).- §9. Hardy’s interpolation formula.- §10. Two inequalities of S. Bernstein.- §11. Several variables.- III. Fourier-Stieltjes transforms (one variable).- §1. Basic properties.- §2. Distribution functions, and characteristic functions.- §3. Positive-definite functions.- §4. A uniqueness theorem.- Notes.- References.