Math 414-501 — Spring 2021

Test 1 Review

General Information

Time and date. Test 1 will be given on Friday, 2/26/21, at 1:35, in our usual classroom.

Bluebooks. Please bring an 8½×11 bluebook.

Calculators. You may use calculators to do arithmetic, although you will not need them. You may not use any calculator that has the capability of doing linear algebra or storing programs or other material.

Other devices. You may not use cell phones, computers, or any other device capable of storing, sending, or receiving information.

Structure and coverage. There will be 5 to 7 questions, some with multiple parts. The test will cover sections 1.2.1-1.3.5 in the text, the sine and cosine series, and the notes on point-wise convergence of Fourier series. The problems will be similar to ones done for homework, and as examples in class and in the text. A short table of integrals will be provided. Here are links to practice tests: 2002, 2003 and 2009. Be aware that these tests cover some material that will not be on the test for this class.

Topics Covered

Calculating Fourier Series
  1. Extensions of functions — periodic, even periodic, and odd periodic extensions. Be able to sketch extensions of functions. 1/25/21, 1/27/21
  2. Fourier series. Be able to compute Fourier series in either real or complex forms, and with prescribed period $2\pi$ on an intervals of the form $[-\pi,\pi]$, $[0, 2\pi]$. You may be asked to use other intervals, $[-a,a]$ for some value of $a$. (I'll make it clear to you if $a\ne \pi$.)
  3. Fourier sine series (FSS) and Fourier cosine series (FCS). Be able to compute FSS and FCS for functions defined on a half interval. §1.2.3, the notes on sine and cosine series 1/27/21

Pointwise convergence

  1. Definition of pointwise convergence.
  2. Definitions of piecewise continuous, jump discontinuity, and piecewise smooth.
  3. Proof of pointwise convegence. The material below, which was covered in the classes on 2/5, 2/8 and 2/10, comes from the notes on point-wise convergence of Fourier series.
    1. Riemann-Lebesgue Lemma. Be able to give a proof of this in the simple case that $f$ is continuously differentiable.
    2. Fourier (Dirichlet) kernel, $P_N$. Know how to express partial sums in terms of $P_N$.
    3. Be able to sketch a proof for pointwise convergence of a FS, making use of the formula for $P_N$ and the properties of $P_N$ as well as the Riemann-Lesbegue Lemma. (To sketch a proof means to list its principle parts, but without details of their proofs.)
  4. Be able to use the theorems on pointwise convergence (Theorem 1.22 & 1.28) to decide the function that an FS, FSS, or FCS converges to.
  5. Be able to use pointwise convergence to sum a series. See the examples from class and in the notes mentioned above. 1/27/21

Uniform convergence

  1. Definition of uniform convergence.
  2. Be able to prove Theorem 1.30, under the condition that $f$ is twice continuously differentiable.
  3. Know the conditions under which an FS, FSS, or FCS is uniformly convergent. Be able to apply these to determine whether or not an FS is uniformly convergent. §1.3.4, 2/12/21.
  4. Gibbs' phenomenon. Be able to briefly describe the Gibbs' phenomenon. Problem 1.32, 2/12/21

Mean convergence

  1. Be able to state the definition of mean convergence.
  2. Parseval's equation. Know both the real and complex form, Theorems 1.39 and 1.40. Be able to use this theorem to sum series. See Example 1.4.1.

Updated 2/21/2021.