Mathematics

For graduate-level courses in Fourier or Harmonic Analysis.

Designed specifically for students (rather than researchers), this introduction to Fourier Analysis starts where the real and complex first-year graduate classes end. Students gain a solid foundation in basic concepts through detailed, user-friendly explanations, acquire deeper understanding by working through a variety of examples and exercises, and broaden their applied perspective by reading about recent developments and advances in the subject.

• This text offers a unique, broad overview of the subject.

  • Provides both a link to the roots of Fourier analysis as well as to the most current developments in the field.

• Over 500 exercises with hints—Range from simple calculations to challenging problems.

  • Enriches the material and provides students with opportunities to develop additional mathematical intuition and deeper comprehension.

• Detailed and careful presentation of the material—With several examples worked out completely.

  • Facilitates the understanding of first-time readers and teaches students how to do mathematics rigorously.

• A detailed proof of the Carleson-Hunt theorem on almost everywhere convergence of Fourier series and integrals of L ^p functions is included—One of the most difficult and celebrated theorems in Fourier Analysis.

  • Both instructors and students will clearly profit from the careful presentation of this theorem.

• Extensive historical notes.

  • Provides accurate accounts of past research and suggests directions for future investigation.

• Complete Appendix—Contains a variety of miscellaneous formulae needed in auxiliary operations that appear in the text.

  • Provides students with one-source convenience—no need to refer to other books.

Prolegomena.


 1. L ^p Spaces and Interpolation.


 2. Maximal Functions, Fourier Transform, and Distributions.


 3. Fourier Analysis on the Torus.


 4. Singular Integrals of Convolution Type.


 5. Littlewood-Paley Theory and Multipliers.


 6. Smoothness and Function Spaces.


 7. BMO and Carleson Measures.


 8. Singular Integrals of Nonconvolution Type.


 9. Weighted Inequalities.


10. Boundedness and Convergence of Fourier Integrals.


Bibliography.


Index of Notation.


Index.

Harmonic Analysis  (Mathematics)
Fourier Analysis  (Mathematics)

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Loukas Grafakos is a native of Athens, Greece. He earned his doctoral degree at UCLA and is currently a Professor of Mathematics at the University of Missouri. He has taught at Yale University and Washington University in St. Louis and he has also held visiting positions at the Mathematical Sciences Research Institute in Berkeley and the University of Pittsburgh. He has been named a Kemper Fellow for Excellence in Teaching and he has authored or co-authored over forty research articles in Fourier analysis. An avid traveler, he has visited over one hundred countries and has given many international lectures.

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