MATH 712 – Homework

Homework Assignments

04/28
  1. Continue reading the notes on primitives of distributions. Send me a question
    by Monday evening.
    Due 05/02.
04/26
  1. Start reading this set of notes on primitives of distributions, including elementary
    vector calculus with distributions. We will discuss this set of notes,
    and Folland’s Exercise 11 from Section 9.1,
    on Thursday.
04/21
  1. Your final exam consists of four exercises from Folland:
    8.22 (page 256); 8.44 (page 276); 8.46 (page 277); and
    EITHER 9.14 (page 291) OR 9.27 (page 300). (Exclusive OR.)
    Due 05/17.
  2. We will discuss proofs from Section 9.2 on Tuesday, so bring questions!
04/19
  1. Finish (re-)reading Section 9.2 (Compactly supported, tempered,
    and periodic distributions) in Folland.
  2. Exercises 9.1: 9, 10, 11, 12; Exercises 9.2: 16, 17, (18, 19, 20, 21).
04/14
  1. Re-read as much of Section 9.1 (Distributions) in Folland as you think necessary,
    in light of today’s discussion. Feel free to send me questions on anything in this
    section that is still not clear to you.
  2. Read Section 9.2 (Compactly supported, tempered, and periodic distributions)
    in Folland, and send me a question by Monday evening. You may omit the last page
    (periodic distributions) if you wish, and you can give a light treatment to compactly-supported
    distributions: most of our effort in this section will be on tempered distributions and the
    Fourier transform.
    Due 04/18.
  3. Hand in Exercises 3.5: 34 (page 108); Exercises 9.1: 7. (You should find the first
    of these to be useful to you in answering the second.)
    Due 04/21.
04/12
  1. Finish reading Section 9.1 (Distributions) in Folland.
  2. Update, 04/13. Here are solutions
    to the midterm exam    . Please let me know of any typos, other errors,
    or necessary clarifications.
04/07
  1. Read as much as you can of Section 9.1 (Distributions) in Folland,
    and send me a question by Monday evening. (I would like us to talk about
    convolutions on Tuesday, so you should try to read at least that far in the
    section.)
    Due 04/11.
  2. Exercises 9.1: 1 through 7.
04/05
  1. Read Section 8.6 (Fourier analysis of measures) in Folland.
  2. Exercises 8.6: 38, 40, 41, 42.
03/31
  1. Read Section 8.5 (Pointwise convergence of Fourier series) in Folland, and send me
    a question by Monday evening.
    Due 04/04.
  2. Exercises 8.5: 33, 34, (35, 37).
03/29
  1. Read Section 8.4 (Summation of Fourier integrals and series) in Folland, at least through
    Theorems 8.35 and 8.36. We will discuss Folland’s “specific examples”, which start on page 260,
    on Thursday; you can read ahead through those, if you feel you need to do so to prepare for the
    discussion.
  2. Exercises 8.4: 24, 26, 27, 29, (32).
  3. Hand in Exercises 8.3: 18; Exercises 8.4: 26.
    Due 04/05.
03/24
  1. Read Section 8.3 (The Fourier transform) in Folland. You should ignore any
    references to the Riesz-Thorin interpolation theorem (unless you have the time to
    go back and read the statement and proof of that theorem in Section 6.5.)
  2. Exercises 8.3: 12, 13, (14,) 16, 18, 23. (Exercise 23 is important; be sure you read it carefully and
    understand it, even if you don’t work it through completely.)
03/08
  1. Read Section 8.2 (Convolutions) in Folland.
  2. (Exercises 8.2: 5, 6, 7, 8, 9, 11.)
03/03
  1. Read Section 8.1 (Preliminaries) in Folland.
    I have also posted a set of notes on the Fourier transform
    (on the Notes page), which you may find
    helpful if you have never seen this material before.
    Send me a question by Monday evening.
    Due 03/07.
  2. Exercises 8.1: 1, 2, (3). (If you have never done exercise 3—or
    an equivalent result—before, you should do it now.)
  3. Start working on the midterm
    exam    .
    Due 03/24.
03/01
  1. Re-read Section 5.4 (Topological vector spaces) in Folland, and send me a question by
    Wednesday evening.
    Due 03/02.
02/25
  1. Read Section 5.4 (Topological vector spaces) in Folland.
  2. Exercises 5.4: 43, 45, 47, (48, 49, 50, 52).
  3. I have posted a proof of the Uniform Boundedness Principle on the
    Notes page. We will use this theorem
    on Tuesday to complete the solution of Folland’s Exercise 7.22. Read the whole proof
    if you have time; otherwise just read the statement of the theorem.
02/23
  1. Read Section 4.3 (Nets) in Folland.
  2. Exercises 4.3: 30, 31, 32, 33, 34, (36).
02/18
  1. Read Section 7.3 (The dual of
    C0(X)) in Folland.
  2. Exercises 7.3: 16, (17), 18, 20, 22, 24, (27).
02/16
  1. Read Section 7.2 (Regularity and approximation theorems)
    in Folland.
  2. Exercises 7.1: 1, 2, (3), 4, (5, 6).
  3. Exercises 7.2: 7, 8, (9), 11, 12, 13, (14, 15). (Exercise 15 is a
    famous and important example, but you will need to know something
    about uncountable ordinals to complete it.)
  4. Hand in Exercises 7.1: 4; Exercises 7.2: 11.
    Due 02/23.
02/11
  1. Read Lemma 4.14, Theorem 4.15 (Urysohn’s Lemma), and
    Section 4.5 (Locally compact Hausdorff spaces)
    in Folland, paying particular attention to the results needed
    for the Riesz representation theorem in Section 7.1.
    (Let me know if we need to discuss any earlier material from
    Chapter 4.)
  2. Exercises 4.5: 47 (you can consider only the LCH case), 51, (52, 54, 55) 56, 57.
  3. Hand in Exercises 4.5: 56.
    Due 02/18.
02/09
  1. Read as much as you can of Section 7.1 (Positive linear
    functionals on Cc(X)) in Folland.
    You will see references to Section 4.5 (Locally compact
    Hausdorff spaces); you should read the relevant theorem
    statements and try to see how they imply the results in
    Section 7.1, but you do not need to study the proofs in Section 4.5
    at this point. (You will read that section next week.)
    Send me a question by Wednesday evening.
    Due 02/10.
02/04
  1. Read Section 6.3 (Some Useful Inequalities)
    in Folland. (You can concentrate on Chebyshev’s Inequality and
    Minkowski’s Inequality for Integrals on a first reading.)
  2. Exercises 6.3: 26, 27, 31, 32.
02/02
  1. Read Section 6.2 (The dual of Lp)
    in Folland. For the case p = 1, the standard theorem is that the
    dual of L1 is isomorphic to L∞
    for σ-finite measures. It would be sufficient to work through the proof
    in this case, although Folland does give some interesting
    generalizations.

  2. Exercises 6.2: 17, 18, 20, 22, (23, 24). (Exercises in parentheses
    are optional.)
  3. Hand in Exercises 6.1: 9, 10; Exercises 6.2: 17.
    Due 02/09.
01/28
  1. Discuss Section 5.5 in class as needed. (If there are questions you cannot answer
    by yourselves, prepare questions for us to discuss next Tuesday.)
  2. Discuss the class exercise set on “Weak Derivatives in One Dimension” (on the “Notes” page
    of the class website). Try to get a complete answer to Question 1, then answer as many parts of
    Question 2 as you can.
  3. Read Section 6.1 (Basic theory of Lp spaces) in Folland.
    Send me a question by Monday evening.
    Due 02/01.
  4. Exercises 6.1: Exercises 1, 2, (3–8), 9, 10, (11, 12, 13, 15).
    (Exercises are in parentheses are interesting but optional.)
01/26
  1. Read Section 5.5 (Hilbert spaces) in Folland.
  2. Exercises 5.5: Exercises (54), 55, (57, 58, 60), 61, 62, (63, 64, 65, 66).
    (Exercises are in parentheses are interesting but optional.)
  3. If you want something of a challenge, show that a norm in a
    vector space can be derived from an inner product if and only if it
    satisfies the parallelogram law (5.22, on page 173). (Hint:
    the polarization identity in Exercise 55a tells you what the only possible
    inner product can be, if the norm is given. The real version of this result
    is somewhat easier to write down than the complex one, but involves
    essentially the same ideas, so it would be sufficient to prove that case.)