# Test 1 Review

## General Information

Time and date. Test 1 will be given on Monday, 2/19/18, 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 (2/15/18), 1-2:30, on Friday (2/16/18), 2:30-3:30.

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 0.5.1 and 1.2.1-1.3.5 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 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 three periods of periodic 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 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

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. (We did the more general piecewise smooth case in class on 2/2/18.) §1.3.1.
4. Fourier (Dirichlet) kernel, $P_N$. Know what $P_N$ is and be able to express $S_N$ in terms of $P_N$. §1.3.2
5. 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. §1.3.1-1.3.2. (To sketch a proof means to list its principle parts.)
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.
7. Be able to use pointwise convergence to sum a series.

Uniform convergence

1. Definition of uniform convergence.
2. Conditions under which an FS, FSS, or FCS is uniformly convergent — Theorem 1.30 (on the function) Lemma 1.33 (on the coefficients), and Be able to apply these to determine whether or not an FS is uniformly convergent. These are all stated for periodic functions. §1.3.4.

Mean convergence

1. Know the definitions of an orthogonal set of vectors, an orthonormal set of vectors, orthogonal projection. Be able to use Theorem 0.21 to find the orthogonal projection of a vector onto a subspace. (See Assignment 3, problem 4.)
2. Be able to state definition of mean convergence.
3. Mean convergence.. Let $f\in L^2[-\pi,\pi]$ (or $L^2[0,2\pi]$). Then the partial sums $S_N$ for $f$ converge in the mean to $f$, and Parseval's Equation holds.
4. Parseval's theorem. Know both the real (Eqn. 1.40) and complex form (Eqn. 1.41). Be able to use this theorem to sum series. See Example 1.4.1.

Updated 2/15/2018.