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Course
Description: We will examine some of the basic objects and
constructions in Harmonic Analysis, such as Fourier Transforms, maximal
functions, and integral and differential operators. A recurrent theme
throughout the course will be the treatment of integrals which are not
absolutely convergent by means of cancellation or oscillatory estimates.
Specific topics are likely to include:
Definition/mapping properties of the Fourier transform in R^n.
- Convergence of Fourier series in 1
dimension. Non-convergence of Fourier series in higher dimensions.
- Hardy-Littlewood maximal function and
singular integrals.
Boundedness of singular integrals over weighted norm spaces.
- Stein-Tomas theorem and other Fourier
restriction results.
- Other topics as time permits. This may
include stationary phase, introduction to pseudo-differential operators,
or Haar functions/wavelet bases.
Grading policy:
Students taking this course for a grade will be asked to complete
a reasonable number of exercises and/or a literature review project. A list
of recommended topics will be provided by the instructor, however you are
free to follow your own interests.
Text: There is no specific textbook for
this course. Lectures will cover selected material from the following
sources:
- Lectures on Harmonic Analysis, by Thomas
Wolff (AMS University Lecture
Series)
- Fourier Analysis, by Javier
Duoandikoetxea (AMS Graduate Studies in
Mathematics)
- Harmonic Analysis: Real-Variable
Methods, Orthogonality, and Oscillatory Integrals, by Elias Stein
(Princeton University Press)
- An introduction to Harmonic Analysis, by
Yitzhak Katznelson (Dover
Publications)
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