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This contemporary graduate-level text in harmonic analysis introduces the reader to a wide array of analytical results and techniques.
'This two-volume text in harmonic analysis introduces a wealth of analytical results and techniques. It is largely self-contained and will be useful to graduate students and researchers in both pure and applied analysis. Numerous exercises and problems make the text suitable for self-study and the classroom alike. This first volume starts with classical one-dimensional topics: Fourier series; harmonic functions; Hilbert transform. Then the higher-dimensional Calderón Zygmund and Littlewood Paley theories are developed. Probabilistic methods and their applications are discussed, as are applications of harmonic analysis to partial differential equations. The volume concludes with an introduction to the Weyl calculus. The second volume goes beyond the classical to the highly contemporary and focuses on multilinear aspects of harmonic analysis: the bilinear Hilbert transform; Coifman Meyer theory; Carleson's resolution of the Lusin conjecture; Calderón's commutators and theCauchy integral on Lipschitz curves. The material in this volume has not previously appeared together in book form.This two-volume text in harmonic analysis is appropriate for advanced undergraduate students with a strong background in mathematical analysis and for beginning graduate students wishing to specialize in analysis. With numerous exercises and problems it is suitable for independent study as well as for use as a course text.
Preface; Acknowledgements; 1. Fourier series: convergence and summability; 2. Harmonic functions, Poisson kernel; 3. Conjugate harmonic functions, Hilbert transform; 4. The Fourier Transform on Rd and on LCA groups; 5. Introduction to probability theory; 6. Fourier series and randomness; 7. Calderón Zygmund theory of singular integrals; 8. Littlewood Paley theory; 9. Almost orthogonality; 10. The uncertainty principle; 11. Fourier restriction and applications; 12. Introduction to the Weyl calculus; References; Index.