This self-contained text introduces Euclidean Fourier Analysis to graduate students who have completed courses in Real Analysis and Complex Variables. It provides sufficient content for a two course sequence in Fourier Analysis or Harmonic Analysis at the graduate level. In true pedagogical spirit, each chapter presents a valuable selection of exercises with targeted hints that will assist the reader in the development of research skills. Proofs are presented with care and attention to detail. Examples are provided to enrich understanding and improve overall comprehension of the material. Carefully drawn illustrations build intuition in the proofs. Appendices contain background material for those that need to review key concepts.
Compared with the author’s other GTM volumes (
Classical Fourier Analysis and
Modern Fourier Analysis), this text offers a more classroom-friendly approach as it contains shorter sections, more refined proofs, and a wider range of exercises. Topics include the Fourier Transform, Multipliers, Singular Integrals, Littlewood–Paley Theory, BMO, Hardy Spaces, and Weighted Estimates, and can be easily covered within two semesters.
Arvustused
This book provides an introduction to Fourier analysis on Euclidean spaces intended for students who have completed first-year graduate courses in real and complex analysis. The text is self-contained and complete with numerous exercises in each section and seven appendices. (Cody B. Stockdale, Mathematical Reviews, May, 2025)
The well-written monograph is intended to serve the purposes of a two-semester course. ... this textbook is very useful for graduate students in mathematics and a convenient reference for researchers working on multi-dimensional Fourier analysis. (Manfred Tasche, zbMATH 1551.42001, 2025)
1 Introductory Material.- 2 Fourier Transforms, Tempered Distributions,
Approximate Identities.- 3 Singular Integrals.- 4 Vector-Valued Singular
Integrals and LittlewoodPaley Theory.- 5 Fractional Integrability or
Differentiability and Multiplier Theorems.- 6 Bounded Mean Oscillation.- 7
Hardy Spaces.- 8 Weighted Inequalities.- Historical Notes.- Appendix A
Orthogonal Matrices.- Appendix B Subharmonic Functions.- Appendix C Poisson
Kernel on the Unit Strip.- Appendix D Density for Subadditive Operators.-
Appendix E Transposes and Adjoints of Linear Operators.- Appendix F Faa di
Bruno Formula.- Appendix G Besicovitch Covering Lemma.- Glossary.-
References.- Index.
Loukas Grafakos is the Mahala and Rose Houchins Distinguished Professor of Mathematics at the University of Missouri at Columbia. He is author of 3 Graduate Texts in Mathematics: Classical Fourier Analysis (GTM 249), Modern Fourier Analysis (GTM 250), and Fundamentals of Fourier Analysis (GTM 302). His research is in Harmonic Analysis.
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