Math 428: Introduction to Fourier Analysis


Course Information

Professor: Kiril Datchev
Email: kdatchev@purdue.edu
Lectures: Mondays, Wednesdays, and Fridays 8:30 to 9:20 am, in STON 217.
Office hours: Mondays, Wednesdays, and Fridays 10:20 to 11:10 in STON 217 and Tuesdays 11:10 to 12:00 in REC 122, or by appointment.

Textbook: An Introduction to Fourier Series and Integrals, by Robert T. Seeley. See also these notes.

The topics we will cover include Fourier series, convolutions, kernels, summation methods, and Fourier transforms, with applications to the wave, heat, and Laplace equations.

Grading will be based on

  • Almost weekly homework assignments, worth 20% of the total grade,
  • two in-class midterm exams, worth 40% of the total grade,
  • a final exam, from 7 to 9 pm in 217 STON on Tuesday May 4th, worth 40% of the total grade.

    • If any problem, such as illness, quarantine, or anything else, interferes with your ability to attend class and do the required work, please email me so that we can arrange a suitable accomodation.

    Homework

    Homework is due on paper at the beginning of class on Wednesdays. Here are the assignments and associated handouts:

    Homework 1, due January 27th.
    Homework 2, due February 3rd. See also this handout from Spivak on uniform convergence.
    Homework 3, due February 10th.
    There is no homework due February 17th, but the first midterm is on Monday, February 15th.
    Homework 4, due February 24th.
    Homework 5, due March 3rd.
    Homework 6, due March 10th.
    Homework 7, due March 17th.
    Homework 8, due March 24th.
    There is no homework due March 31st or April 7th, but the second midterm is on Monday, April 5th.
    Homework 9, due April 14th.
    Homework 10, due April 21st.
    There is no homework due April 28th, but the final is on Tuesday, May 4th.


    Additional Resources

    Our textbook is quite short. Below are some books recommended for further reading.

    Fourier Analysis, by T. W. Körner, is 'a series of interlinked essays', with many many examples and applications. Seeley's and Körner's books both avoid the more advanced theory of Lebesgue integration. An introduction to this theory, with an emphasis on applications to Fourier analysis, is given in Sz.-Nagy's Introduction to real functions and orthogonal expansions. For more on Fourier series, see Hardy and Rogosinski's Fourier Series. For more on the Fourier transform, including applications to differential equations, see Friedlander and Joshi's Introduction to the Theory of Distributions. A more down-to-earth introduction to distributions, with an emphasis on computations and examples, is Kanwal's Generalized Functions: Theory and Applications.

    For more general background and inspiration, I recommend Riemann's habilitation thesis (which is where, incidentally, the Riemann integral was first defined), and which can be found in this beautiful edition of his Collected Papers; see also this handout.




    Finally, a list of general policies and procedures can be found here.