# Test 1 Review

## General Information

Time and date. Test 1 will be given on Wednesday, 2/17/16, at 10:20, in our usual classroom.

Bluebooks. Please bring an 8½×11 bluebook.

Office hours. I will have office hours on Monday (2/15/16), 1:40-4, and on Tuesday (2/16/15), 11:30-1:30.

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 4 to 6 questions, some with multiple parts. The test will cover sections 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. We will only deal with 2π periodic functions.
2. Extensions of functions — periodic, even periodic, and odd periodic extensions. Be able to sketch extensions of functions.
3. Fourier series. Be able to compute Fourier series in either real or complex forms, and with prescribed period 2π on an intervals of the form [−π, π], [0, 2π], or [c − π, c + π].
4. Fourier sine series (FS for odd, 2π-periodic extension) and Fourier cosine series (FS for even, 2π-periodic extension). Be able to compute FSS and FCS for functions defined on a half interval, [0,π].

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. §1.3.1.
4. Fourier (Dirichlet) kernel, P. Know what P is and how to express partial sums in terms of P. §1.3.2
5. Be able to sketch a proof for pointwise convergence of a FS, making use of the formula for P and the properties of P as well as the Riemann-Lesbegue Lemma. §1.3.1-1.3.2. (To sketch a proof means to list its principle parts. We did this in class: see the notes for 2/1/16-2/5/16.)
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. See Exercise 21(b).

Uniform convergence

1. Definition of uniform convergence.
2. 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. These are all stated for periodic functions. To apply them on [−π,π], or [0,π], work with the appropriate periodic extension. §1.3.4.
3. Gibbs' phenomenon. Be able to briefly describe the Gibbs' phenomenon.

Mean convergence

1. Definition of mean convergence.
2. Mean approximation property of partial sums. (See notes for 2/12/16). Be able to show that if f is in L2, with complex Fourier coefficients αn's, then $\int_{-\pi}^\pi |f(x)-S_N(x)|^2dx = \int_{-\pi}^\pi |f(x)|^2dx - 2\pi \sum_{n=-N}^N |\alpha_n|^2.$
3. Parseval's theorem. Know both the real and complex form. Be able to use this theorem to sum series. See Example 1.4.1.
4. Here are the main theorems on mean convergence.
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 2/14/2016.