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Time and place: Monday 2-3 in B3.03 and Tuesday 5-7pm in A1.01. Supervisions Tuesday 4-5 in C1.06 (17 January) and L5 (other days) by Stephen Tate.

Assessment: 3-hour examination.

Description:
Fourier series and Fourier transform have numerous applications in PDEs, functional analysis, probability theory, and even number theory! We will study their definitions and properties, and consider specific applications.

We will mainly follow the book of Stein and Shakarchi (see the references below), with a few extra topics such as distributions, the Poisson equation, the Laplace-Beltrami operator, and Peter-Weyl theorem. There should be lecture notes for the extra topics.

I occasionally put comments on my facebook page; but anything of importance can also be found here.

Assignments:
Assignment 1: Exercises 2, 4, 6, 7, 8 on pages 59-61.
Assignment 2: Exercises 12, 13, 14, 19 on pages 62-64.
Assignment 3: Exercises 12, 13, 14, 15 on pages 91-92.
Assignment 4: Exercises 11, 12, 13, 15, 23 on pages 164-169.
Assignment 5: Exercises 4, 7, 8, and Problem 3, on pages 208-214.
Assignment 6: Exercises 5, 6, 10 on pages 209-211.

References:

The main reference is Fourier Analysis of E.M. Stein and R. Shakarchi, Princeton, 2003. The assignments above refer to this book. But the following references are very much worth studying:

  • G. Frieseke, Lectures on Fourier Analysis, 2007.

  • A. Pinkus and S. Zafrany, Fourier Series and Integral Transforms, Cambridge, 1997.

  • G.B. Folland, Real Analysis, Wiley, 1999.

  • E.H. Lieb and M. Loss, Analysis, AMS, 2001.

  • J.K. Hunter and B. Nachtergaele, Applied Analysis, World Scientific, 2001. (This book is available online.)

  • M. Reed and B. Simon, Fourier Analysis, Self-Adjointness (Methods of Modern Mathematical Physics II), Elsevier, 1975.