Math 172: Lebesgue Integration and Fourier Analysis

Winter quarter 2011/2012

Instructor: Richard Bamler

  • office 382-N
  • rbamler@math.stanford.edu
  • 650-723-2975
  • office hours: Mon 1:00 pm - 3:00 pm, Thu 1:00 pm - 2:00 pm or by appointment

    • Course Assistant: Jeremy Leach

  • office 381-J
  • jleach "a t" math.stanford.edu
  • office hours: Thur 3:45 pm - 6:45 pm
    • Course times: MoWeFr 10:00 am - 10:50 am
    Room: Building 380, Room 380-F
    Course book: F. Jones, "Lebesgue Integration on Euclidean Spaces", Jones and Bartlett Mathematics (2001)

    THE FINAL EXAM WILL BE ON WEDNESDAY 3/21, 8:30 AM - 11:30 AM AT 380-F (THE CLASS ROOM).
    SOLUTION TO THE FINAL

    It will be closed book and take about 2.5h.

    FAQ

  • How much will the exam focus on the Fourier Transform, how much do I have to study for it?
    The Fourier transform was more or less additional material. There will be a problem on the FT, but you should be able to solve it if you understand the basic principles. So when you study for the exam, make sure that you understand the material from class. Your time is invested best if you focus your studies on the other topics.
  • Will the exam be as hard as the problem sets?
    No! The purpose of the problem sets was to develop material which would have been to technical or to long to carry out class. Some problems also provided more advanced material which was on the syllabus of this class, but which was not really essential for keeping up with class. The final will consist of much easier problems that you can write up in a small paragraph once you got the main idea. This does not mean that material from the problem sets will not be touched in the exam.
  • Will the final be like the midterm?
    Yes! It will just be 2.5 times as long since there is 2.5 times as much time and it will cover both the material from before and after the midterm.
  • How should I study for the final?
    The best thing is to go through your notes and try to replicate the proof of every Lemma and Theorem by yourself. Then do the same thing with the problem sets.

    • Organization of the course

  • Homeworks will be assigned weekly, they can be downloaded from the course website and are due each Monday, unless otherwise specified
  • There will be one midterm, on Monday, 02/13/12 from 4:00pm to 5:30pm (room 380-F), the makeup exam is on Wednesday 02/15/12 (tentative date). Students are only eligible to attend the makeup exam if there is a justified reason that they could not attend the main exam (e.g. illness, religious holiday, clash with an evening class/official Stanford event).
  • The final will be on Wednesday, 03/21/12, 8:30 am - 11:30 am
  • The final grade will be based on the homework assignments (30%), the midterm (30%) and the final (40%)

    • Manuscript

    There has been demand to make my manuscript public. I was hesitating to do so, since it is actually not ready for publication and should not be used as a replacement for your own notes. As a compromise, I will post completed chapters from a while ago. Please note that there might be a lot of typos and mistakes in this manuscript. I correct them on the blackboard, but don't update them in the manuscript. I also don't stick completely to these notes completely (i.e. I sometimes leave out/add material in class). I strongly recommend to use the course book instead to review past classes!

    Chapters I.1-I.2
    Chapter I.3
    Chapter I.4
    Chapter II

    Schedule

    The following schedule is subject to change!

    Week 1: 01/09 - 01/13

  • Class 1 (Mon): Overview
  • Class 2 (Wed): Overview (Remark on I.2.3 Topology)
  • Class 3 (Fri): Overview / Lebesgue Measure
    • Problem Set 1 (due Wed 1/18) (solution)

    Week 2: 01/16 - 01/20

  • Class 4 (Wed): Lebesgue Measure
  • Class 5 (Fri): Lebesgue Measure
    • Problem Set 2 (due Mon 1/23 in class) (solution)

    Week 3: 01/23 - 01/27

  • Class 6 (Mon): Lebesgue Measure
  • Class 7 (Wed): Lebesgue Measure
  • Class 8 (Fri): Lebesgue Measure / Invariance of the Lebesgue Measure
    • Problem Set 3 (due Mon 1/30 in class) (solution)

    Week 4: 01/30 - 02/03

  • Class 9 (Mon): Algebras of Sets
  • Class 10 (Wed): Algebras of Sets
  • Class 11 (Fri): Algebras of Sets / Measurable Functions
    • Problem Set 4 (due Mon 2/6 in class) (solution)

    Week 5: 02/06 - 02/10

  • Class 12 (Mon): Measurable Functions Integration / Limits of Integrals
  • Class 13 (Wed): Measurable Functions / Integration
  • Class 14 (Fri): Integration
    • Problem Set 5 (due Fri, 2/17 in class, (note the change)) (solution)

    Week 6: 02/13 - 02/17

  • Class 15 (Mon): Lebesgue Integration; Midterm from 4:00pm to 5:30pm (room 380-F) (solution)
  • Class 16 (Wed): Lebesgue Integration; makeup Midterm in the afternoon
  • Class 17 (Fri): Lebesgue Integration
    • Problem Set 6 (due Wed 2/22 in class) (solution)

    Week 7: 02/20 - 02/24

  • Class 18 (Wed): Lebesgue Integration / $L^p$-Spaces
  • Class 19 (Fri): $L^p$-Spaces
    • Problem Set 7 (due Mon 3/5 in class, Note the change) (solution)

    Week 8: 02/27 - 03/02

  • Class 20 (Mon): $L^p$-Spaces
  • Class 21 (Wed): $L^p$-Spaces / Transformation law
  • Class 22 (Fri): Transformation law
    • Remark on bump functions

    Week 9: 03/05 - 03/09

  • Class 23 (Mon): Transformation law / Fubini's Theorem
  • Class 24 (Wed): Fubini's Theorem
  • Class 25 (Fri): Convolution / Fourier Transform
    • Problem Set 8 (due Mon 3/12 in class, Note: I made some major changes!!) It's also okay if you submit the problem set on Wednesday (solution)

    Week 10: 03/12 - 03/16

  • Class 26 (Mon): Fourier Transform
  • Class 27 (Wed): Fourier Transform
  • Class 28 (Fri): Fourier Transform
    • This Java applet is a nice visualization for Fourier series. It might help you to get a feel for the Fourier Transform. Fourier series are similar to the Fourier transform: The difference is that the function f is not in L^1 but periodic and integrable on one period. The transform of it is not a function, but a discrete sequence of complex numbers, called Fourier weights (the formula for computing the weights is almost the same as the one for computing the Fourier Transform, also there is a similar formula for computing a periodic function from the Fourier weights). In the applet, the weights are represented in a little different way, but the effect is essentially the same (you might want to check "Mag/Phase View"). Try the following:
  • Draw a periodic function in the top field with your mouse cursor. Make sure that you jiggle enough to create noise. The Fourier weights are computed immediately, by integrating against a wave function.
  • Adjust the weights, the corresponding wave function is displayed in yellow. The applet computes the corresponding periodic function using the integral that we discussed in class. If you drew this new periodic function on top, then the weights would turn out to be exactly the same as the the ones you entered. That's the content of the Fourier Inversion Theorem in this setting.
  • Set the weights for the high frequencies to zero. Observe that the noise disappears.
  • Play the sound for these examples (press the sound checkbox, you might have to adjust "Playing Frequency"). Which graph describes better what you hear, the top one or the lower two?
  • Draw a highly oscillating function on the top (i.e. a very sloppy sine function). How do you determine its average frequency?

    • Final exam: 03/21, 8:30 am - 11:30 am

    (solution)
    last modified: 03/14/12