Classical Fourier Transforms by Komaravolu ChandrasekharanClassical Fourier Transforms by Komaravolu Chandrasekharan

Classical Fourier Transforms

byKomaravolu Chandrasekharan

Paperback | December 22, 1988

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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.
Title:Classical Fourier TransformsFormat:PaperbackDimensions:179 pagesPublished:December 22, 1988Publisher:Springer Berlin HeidelbergLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3540502483

ISBN - 13:9783540502487


Table of Contents

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.