Contents

I Fourier Series on T 11 Fourier coefficients . . . . . . . . . . . . . . . . . . . . . 22 Summability in norm and homogeneous banach

spaces on T . . . . . . . . . . . . . . . . . . . . . . . . . 93 Pointwise convergence of σn(f). . . . . . . . . . . . . . . 184 The order of magnitude of Fourier coefficients . . . . . . 235 Fourier series of square summable functions . . . . . . . 296 Absolutely convergent Fourier series . . . . . . . . . . . 337 Fourier coefficients of linear functionals . . . . . . . . . 378 Additional comments and applications . . . . . . . . . . 51

�9 The d-dimensional torus . . . . . . . . . . . . . . . . . 63

II The Convergence of Fourier Series 671 Convergence in norm . . . . . . . . . . . . . . . . . . . . 672 Convergence and divergence at a point . . . . . . . . . . 72

�3 Sets of divergence . . . . . . . . . . . . . . . . . . . . . . 76

III The Conjugate Function 831 The conjugate function . . . . . . . . . . . . . . . . . . . 832 The maximal function of Hardy and Littlewood . . . . . 963 The Hardy spaces . . . . . . . . . . . . . . . . . . . . . . 105

IV Interpolation of Linear Operators 1171 Interpolation of norms and of linear operators . . . . . . 1172 The theorem of Hausdorff–Young . . . . . . . . . . . . . 1233 Marcinkiewicz’s theorem . . . . . . . . . . . . . . . . . . 128

V Lacunary Series and Quasi-analytic Classes 1331 Lacunary series . . . . . . . . . . . . . . . . . . . . . . . 133

�2 Quasi-analytic classes . . . . . . . . . . . . . . . . . . . . 142

vii

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VIII AN INTRODUCTION TO HARMONIC ANALYSIS

VI Fourier Transforms on the Line 1511 Fourier transforms for L1(R) . . . . . . . . . . . . . . . . 1522 Fourier–Stieltjes transforms . . . . . . . . . . . . . . . . 1633 Fourier transforms in Lp(R), 1 < p ≤ 2 . . . . . . . . . . 1744 Tempered distributions and pseudomeasures . . . . . . . 1815 Almost-Periodic functions on the line . . . . . . . . . . 1916 The weak-star spectrum of bounded functions . . . . . . 2057 The Paley–Wiener theorems . . . . . . . . . . . . . . . . 208

�8 The Fourier–Carleman transform . . . . . . . . . . . . . 2149 Kronecker’s theorem . . . . . . . . . . . . . . . . . . . . 217

VII Fourier Analysis on Locally CompactAbelian Groups 223

1 Locally compact abelian groups . . . . . . . . . . . . . . 2232 The Haar measure . . . . . . . . . . . . . . . . . . . . . 2243 Characters and the dual group . . . . . . . . . . . . . . . 2254 Fourier transforms . . . . . . . . . . . . . . . . . . . . . 2275 Almost-periodic functions and the Bohr

compactification . . . . . . . . . . . . . . . . . . . . . . . 229

VIII Commutative Banach Algebras 2311 Definition, examples, and elementary properties . . . . . 2312 Maximal ideals and multiplicative

linear functionals . . . . . . . . . . . . . . . . . . . . . . 2353 The maximal-ideal space and the

Gelfand representation . . . . . . . . . . . . . . . . . . . 2424 Homomorphisms of Banach algebras . . . . . . . . . . . 2515 Regular algebras . . . . . . . . . . . . . . . . . . . . . . . 2586 Wiener’s general Tauberian theorem . . . . . . . . . . . . 2647 Spectral synthesis in regular algebras . . . . . . . . . . . 2678 Functions that operate in regular

Banach algebras . . . . . . . . . . . . . . . . . . . . . . . 2739 The algebra M(T) and functions that operate on

Fourier–Stieltjes coefficients . . . . . . . . . . . . . . . 28210 The use of tensor products . . . . . . . . . . . . . . . . . 287

A Vector-Valued Functions 2951 Riemann integration . . . . . . . . . . . . . . . . . . . . 2952 Improper integrals . . . . . . . . . . . . . . . . . . . . . . 296

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CONTENTS IX

3 More general integrals . . . . . . . . . . . . . . . . . . . 2964 Holomorphic vector-valued functions . . . . . . . . . . . 296

B Probabilistic Methods 2991 Random series . . . . . . . . . . . . . . . . . . . . . . . . 2992 Fourier coefficients of continuous functions . . . . . . . 3023 Paley–Zygmund,

(when

∑|an|2 = ∞). . . . . . . . . . 303

Bibliography 307

Index 311

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