Math 414-501 — Spring 2022

Test 1 Review

General Information

Time and date. Test 1 will be given on Wednesday, 2/16/2022, at 11:30, in our usual classroom.

Bluebooks. Please bring an 8½×11 bluebook.

Office hours. I will have office hours on Monday (2/14/22), 12:30-1:30 & 2-3, and on Tuesday (2/15/22), 11:30-1:30 and 2-3.

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 1.2.1-1.3.3 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

Fourier Series
  1. Calculating 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 $a\ne \pi$.)
  2. 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
  3. Be able to prove Lemma 1.3 for $2\pi$-periodic functions and to use it to justify the formulas for coefficients on intervals of the form $[0,2\pi]$, as well as $[-\pi,\pi]$.
  4. Extensions of functions. Be able to sketch periodic, even periodic, and odd periodic extensions. Be able to sketch several periods of the function that a Fourier series converges to.
  5. Complex Fourier series. Be able to compute the complex form of a Fourier series. Be able to go from the complex form to a real form or from a real from to a complex one. (§ 1.2.5)

Pointwise convergence

  1. Definition of pointwise convergence.
  2. Definitions of piecewise continuous, jump discontinuity, and piecewise smooth.
  3. Proof of pointwise convegence. See the notes on point-wise convergence of Fourier series.
    1. Dirichlet (Fourier) kernel, $P_N$. Know how to express partial sums $S_N$ in terms of $P_N$.
    2. Be able to show that $P_N(u) = \frac{1}{2\pi} + \frac{1}{\pi}\sum_{n=1}^N \cos(nu)$, and that it is even and $2\pi$-periodic.
    3. Be able to use Lemma 1.3 to show $E_N$ has the form in the notes.
    4. Riemann-Lebesgue Lemma. Be able to give a proof of this in the simple case that $f$ is continuously differentiable. Note: the proof in the book is incomplete. Only the proof given in my notes on pointwise convergence will be accepted.
    5. Be able to finish the proof of pointwise convergence using the results above.
  4. 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 principal parts, as was done above.)
  5. Be able to use the theorems on pointwise convergence (Theorem 1.22 & 1.28) to determine the function that an FS, FSS, or FCS converges to.
  6. Be able to use pointwise convergence to sum a series. See the examples from class and in the notes mentioned above. Examples were given in class, in problems and in my notes.

Updated 2/12/2022.