Table of Contents

1 Fourier Series and Integrals.- 1.1. Definitions and easy results.- 1.2. The Fourier transform.- 1.3. Convolution; approximate identities; Fejér’s theorem.- 1.4. Unicity theorem; Parseval relation; Fourier-Stieltjes coefficients.- 1.5. The classical kernels.- 1.6. Summability; metric theorems.- 1.7. Pointwise summability.- 1.8. Positive definite sequences; Herglotz’ theorem.- 1.9. The inequality of Hausdorff and Young.- 1.10. Multiple Fourier series; Minkowski’s theorem.- 1.11. Measures with bounded powers; homomorphisms of l1.- 2. The Fourier Integral.- 2.1. Introduction.- 2.2. Kernels on R.- 2.3. The Plancherel theorem.- 2.4. Another convergence theorem; the Poisson summation formula.- *2.5. Finite cyclic groups; Gaussian sums.- * Starred sections present material that is less fundamental..- 3. Hardy Spaces.- 3.1. Hp(T).- 3.2. Invariant subspaces, factoring, proof of the theorem of F. and M. Riesz.- 3.3. Theorems of Beurling and Szegö.- 3.4. Structure of inner functions.- 3.5. Theorem of Hardy and Littlewood; Hilbert’s inequality.- 3.6. Hardy spaces on the line.- 4. Conjugate Functions.- 4.1. Conjugate series and functions.- 4.2. Theorems of Kolmogorov and Zygmund.- 4.3. Theorems of M. Riesz and Zygmund.- 4.4. The conjugate function as a singular integral.- 4.5. The Hilbert transform.- 4.6. Maximal functions.- 4.7. Rademacher functions; absolute Fourier multipliers.- 5. Translation.- 5.1. Theorems of Wiener and Beurling; the Titchmarsh convolution theorem.- 5.2. The Tauberian theorem.- 5.3. Spectral sets of bounded functions.- *5.4. A theorem of Szegö; theorem of Gru?ewska and Rajchman; idempotent measures.- 6. Distribution.- 6.1. Equidistribution of sequences.- 6.2. Distribution of (nku).- 6.3. Dynamical systems; (k2u).- Appendix. Integration by parts.-Bibliographic Notes.

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