Math 414-501 — Spring 2019

Test 1 Review

General Information

Time and date. Test 1 will be given on Monday, 2/18/19, at 10:20, in our usual classroom.

Bluebooks. Please bring an 8½×11 bluebook.

Office hours. In addition to my usual office hours, I will have office hours on Thursday (2/14/19), 3-4:30, Friday (2/15/19), and on Monday (2/18/19), 9-9:50.

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 sections 1.2.1-1.3.5 in the text, my notes on Fourier sine and cosine series, pointwise convergence of Fourier series, and least squares and 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.
  2. Fourier series. Be able to compute Fourier series in either real or complex forms, and with prescribed period $2\pi$ on intervals of the form $[-\pi,\pi]$ or $[0, 2\pi]$. (We will not use $a\ne\pi$.)
  3. Fourier sine series (FSS) and Fourier cosine series (FCS). Be able to compute FSS and FCS for functions defined on a half interval.

Pointwise convergence

  1. Definition of pointwise convergence.
  2. Definitions of piecewise continuous, jump discontinuity, and piecewise smooth.
  3. Riemann-Lebesgue Lemma. Be able to give a proof of this in the simple case that f is continuously differentiable. Know the proof in my notes on pointwise convergence of Fourier series.
  4. Fourier (Dirichlet) kernel, $P_N$. Know what $P_N$ is and how to express partial sums in terms of $P_N$. §1.3.2 and my notes on pointwise convergence of Fourier series.
  5. Be able to sketch a proof for pointwise convergence of a FS — i.e., list its principal parts. (We did this in class: see the notes for 2/4/19.)
  6. Be able to use the theorems on pointwise convergence (Theorem 1.22 & 1.28) to decide what function an FS, FSS, or FCS converges to, and to evaluate series. (See Exercise 21(b).)

Uniform convergence

  1. Definition of uniform convergence.
  2. Conditions under which an FS, FSS, or FCS is uniformly convergent. §1.3.4.
  3. Gibbs' phenomenon. Be able to briefly describe the Gibbs' phenomenon.

Mean convergence

  1. Definition of mean convergence.
  2. Minimization property of partial sums. This done in one of two ways. The first is given in my notes on least squares and Fourier series. The second was done in class on Monday, 2/11/19 (see Theorem 0.5.20). These are also done in Assignment 4, problems 1 and 2. §1.3.5.
  3. Parseval's theorem. Know both the real and complex form. Be able to use this theorem to evaluate a series. See Example 1.4.1 and similar problems in the homework.
  4. Here are the main theorems on mean convergence. Be able to state them.
    1. If f is in L2, then the partial sums of the FS for f converge in the mean to f.
    2. If f is in L2 and Parseval's equation holds, then the partial sums of the FS for f converge in the mean to f.

Updated 2/12/2019.