Fourier Series on T
I Fourier coefficients
2 Summability in norm and homogeneous banach spaces on T
3 Pointwise convergence of(f)
4 The order of magnitude of Fourier coefficients
5 Fourier series of square summable functions
6 Absolutely convergent Fourier series
7 Fourier coefficients of linear functionals
8 Additional comments and applications
9 The d-dimensional torus
II The Convergence of Fourier Series
1 Convergence in norm
2 Convergence and divergence at a point
3 Sets of divergence
III The Conjugate Function
l The conjugate function
2 The maximal function of Hardy and Littlewood
3 The Hardy spaces
IV Interpolation of Linear Operators
1 Interpolation of norms and of linear operators
2 The theorem of Hausdorff-Young
3 Marcinkiewicz's theorem
V Lacunary Series and Quasi-analytic Classes
1 Lacunary series
2 Quasi-analytic classes
VI Fourier Transforms on the Line
1 Fourier transforms for LI(R)
2 Fourier-Stieltjes transforms
3 Fourier transforms in LP(R), 1 < p < 2
4 Tempered distributions and pseudomeasures
5 Almost-Periodic functions on the line
6 The weak-star spectrum of bounded functions
7 The Paley-Wiener theorems
8 The Fourier-Carleman transform
9 Kronecker's theorem
VII Fourier Analysis on Locally Compact Abelian Groups
1 Locally compact abelian groups
2 The Haar measure
3 Characters and the dual group
4 Fourier transforms
5 Almost-periodic functions and the Bohr compactification
VIII Commutative Banach Algebras
1 Definition, examples, and elementary properties
2 Maximal ideals and multiplicative linear functionals
3 The maximal-ideal space and the Gelfand representation
4 Homomorphisms of Banach algebras
5 Regular algebras
6 Wiener's general Tauberian theorem
7 Spectral synthesis in regular algebras
8 Functions that operate in regular Banach algebras
9 The algebra M(T) and functions that operate on Fourier-Stieltjes coefficients
10 The use of tensor products
A Vector-Valued Functions
1 Riemann integration
2 Improper integrals
3 More general integrals
4 Holomorohic vector-valued functions
B Probabilistic Methods
1 Random series
2 Fourier coefficients of continuous functions
3 Paley-Zygmund,
Bibligraphy
Index
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