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MATH 756:  Harmonic Analysis (Spring 2024)

 

          Syllabus

         Results on the Convergence of Fourier Series

          References:

         [D] Duoandikoetxea: Fourier Analysis
         [F] Folland, G: Real Analysis
         [SS] Stein, E and Shakarchi, R: Fourier Analysis: An introduction.

        MIDTERM EXAM: Friday March 22
                 
        Class presentations:
  1. Solving the wave equation (including Ex 9 on page 27 and Ex 3 on page 59)
  2. Solving the heat equation on the disc (including Ex 10 and 11 on page 27 and Ex 18 on page 64) 
  3.  A continuous function whose Fourier series is divergent at a point (including Ex 8 on page 60)   
  4. The Isoperimetric Inequality (including Ex 2, 3, 4 and problem 1).
  5. Weyl's Equidistribution Thm (including Ex 5, 7, 10)
  6. The Laplace Equation and the Heat equation (including Problem 1 on page 28, Ex 19-20 on pages 64-65, Problem 2 on page 66, ex 11-13 pages 124-125)
  7. The multiplier for the ball
        Homework:

         HW0 (Due Jan 19)         HW1  (Due Feb 2)         HW2  (Solve #2, 4-6, 8-9)    HW3 (Solve #8, 9, 12, 13,15, 16(a)-(c), 18)      HW4                HW5

        Class Diary:

        Week of Jan 08: PDE motivation for the study of Fourier series. Start Lp spaces ([F] Chapter 6 Section 1)
         Week of Jan 15: [F] Chapter 6, Finish section 1, start section 2
         Week of Jan 22: [F] Section 6.2 (proofs done for sequence spaces), Section 6.3 (Excluding Thm 6.20 and Corol. 6.21), Section 6.4, Section 6.5.
                                     In section 6.5, we did not prove Riesz-Thorin; for Marcinkievicz, we followed the proof from [D] Chapter 2 (a particular case).
          Week of Jan 29: [SS] Chapter 2, Sections 1, 4 and 5
         Week of Feb 5: [SS] Chapter 2, Sections 2,3. Presentation 1. [SS] Chapter 3, Section 1.
        Week of Feb 12: Presentation 2. [SS] Chapter 3, finish Section 1. Presentation 3.
         Week of Feb 19: I will be out of town. Prepare presentations 4, 5 and 6 (to be presented after the spring break)
         Week of Feb 26: Finish [SS] Chapter 3. Start Chapter 5. Meet regarding presentations.
         Week of March 4: Spring Break
         Week of March 11: I will be out of town. Continue to prepare presentations 4, 5 and 6, and Midterm Exam.
         Week of March 18: [SS] Chapter 5. Fourier transform of distributions. Midterm Exam on Friday March 22.
         Week of March 25: Presentation 5.
         April: Finish [SS] Chapter 5. [D] Chapters 2 and 3.