Classical Fourier Transforms (Paperback)

Komaravolu Chandrasekharan

ISBN 10: 3540502483 / ISBN 13: 9783540502487

Editore: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG, Berlin, 1988

Riguardo questo articolo

Paperback. 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 .*.*.*. 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 illustraiton of a Fourier transformation of a function "not" belonging to L1 (-, ) 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. Shipping may be from multiple locations in the US or from the UK, depending on stock availability.

Codice articolo 9783540502487

Dati bibliografici

• Titolo: Classical Fourier Transforms (Paperback)
• Editore: Springer-Verlag Berlin and Heidelberg GmbH & Co. KG, Berlin
• Data di pubblicazione: 1988
• Lingua: Inglese
• ISBN 10: 3540502483
• ISBN 13: 9783540502487
• Rilegatura: Paperback
• Condizione: new

Descrizione articolo

Riassunto

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contenuti

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.