## Classical and Modern Fourier AnalysisAn ideal refresher or introduction to contemporary Fourier Analysis, this book starts from the beginning and assumes no specific background. Readers gain a solid foundation in basic concepts and rigorous mathematics through detailed, user-friendly explanations and worked-out examples, acquire deeper understanding by working through a variety of exercises, and broaden their applied perspective by reading about recent developments and advances in the subject. Features over 550 exercises with hints (ranging from simple calculations to challenging problems), illustrations, and a detailed proof of the Carleson-Hunt theorem on almost everywhere convergence of Fourier series and integrals of "L p" functions--one of the most difficult and celebrated theorems in Fourier Analysis. A complete Appendix contains a variety of miscellaneous formulae. "L p" Spaces and Interpolation. Maximal Functions, Fourier transforms, and Distributions. Fourier Analysis on the Torus. Singular Integrals of Convolution Type. Littlewood-Paley Theory and Multipliers. Smoothness and Function Spaces. "BMO" and Carleson Measures. Singular Integrals of Nonconvolution Type. Weighted Inequalities. Boundedness and Convergence of Fourier Integrals. For mathematicians interested in harmonic analysis. |

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### Contents

Maximal Functions Fourier Transform and Distributions | 77 |

Fourier Analysis on the Torus | 157 |

Singular Integrals of Convolution Type | 247 |

Copyright | |

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BMO function bounded function Carleson measure Cauchy compact support conclude condition consequence constant Cn constant multiple contained continuous function converges Corollary cube Q defined Definition denote disjoint dyadic cubes equal estimate Example Exercise exists a constant fact finite fixed follows Fourier coefficients Fourier series Fourier transform function on Rn functions f G Rn given by convolution Hardy spaces Hardy-Littlewood maximal operator Hilbert transform Hint Holder's inequality implies interpolation interval kernel Lemma linear operator Lipschitz locally integrable function LP boundedness LP norm LP(Rn maps maximal function mean value zero measurable functions measure space multiindices nonnegative observe obtain polynomial proof of Theorem Proposition prove rectangle required conclusion result Riesz S(Rn satisfies Schwartz function sequence side length singular integrals smooth function Sn_1 subset Suppose supremum tempered distribution uniformly valid weak type 1,1 write