Classical Fourier Transforms: Universitext

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.