Math 414-501 — Spring 2023

Test 2 Review

General Information

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 4 to 6 questions, some with multiple parts. The test will cover my notes pointwise convergence of Fourier series and sections 1.3.3-1.3.5, 2.1.2, 2.2.1 in the text. The problems will be similar to ones done for homework, and as examples in class and in the text. A short table of integrals and Fourier transform properties will be provided. Here are links to practice tests: 2004 and 2012.

Topics Covered

Fourier Series

• Pointwise Convergence of Fourier series
  1. Be able state and prove the Riemann-Lebesgue Lemma, as given in my notes on pointwise convergence of Fourier series.
  2. Know the conditions under which the partial sum of an FS, FSS, FCS is pointwise convergent. Be able to use them to decide what function an FS, FSS, or FCS converges to pointwise. Be able to sketch several periods of the limit. §1.3.3
  3. Be able to use the pointwise convergence to sum series.
• Uniform convergence
  1. Be able to define the term uniform convergence.
  2. Know the conditions under which an FS, FSS, or FCS is uniformly convergent, and be able to use them to determine whether or not a series is uniformly convergent.
  3. Be able to briefly describe the Gibbs' phenomenon.
• Mean convergence
  1. Parseval's theorem. Know both the real and complex form. Be able to use it to sum series similar to ones given in the homework.
  2. Mean convergence theorem. The partial sum $S_N$ of a FS for a function $f$ converges in the mean (i.e., in $L^2$) to $f$ if and only if $f\in L^2$. In addition, $S_N$ converges in the mean to $f$ if and only if Parseval's equation holds.

Fourier Transforms

• Computing Fourier transforms & properties. (§2.1-2.2.1) Be able to compute Fourier transforms and inverse Fourier transforms. Be able to establish the simple properties listed in Theorem 2.6 of the text, and know how to use them. (You will be given a table listing these properties plus a few others, so you do not need to memorize them.)

Updated 3/26/2023.