Komaravolu Chandrasekharan

Classical Fourier Transforms

• Broschiertes Buch

Produktbeschreibung

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.

Produktdetails

• Universitext
• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 978-3-540-50248-7
• 1989.
• Seitenzahl: 184
• Erscheinungstermin: 22. Dezember 1988
• Englisch
• Abmessung: 244mm x 170mm x 11mm
• Gewicht: 341g
• ISBN-13: 9783540502487
• ISBN-10: 3540502483
• Artikelnr.: 24625623

Inhaltsangabe

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.Summary of the Contents:
Fourier transforms on L1-theory.- Fourier transforms on L2-theory.- Fourier-Stieltjes transforms (one variable).- Notes.- References.