Classical Fourier Transforms

Inhaltsverzeichnis

• 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.