# Test 1 Review

## General Information

Test 1 will be given on Monday, 2/24/2014. Please bring an 8½×11 bluebook. I will have office hours on Thursday, 11-1 (Other times will be OK, but if I'm busy I'll tell you to come back later.), Friday, 2-4, and on Monday 9-10.

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 0.1-0.5, 1.2, 1.3, 2.1, 2.2.1 and 2.2.2 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: 2002, 2003 and 2009

## Topics Covered

### Inner Product Spaces

• Inner products
1. Definitions of real and complex inner products, examples of inner product spaces.
• Standard inner products on Rn, Cn, L2 and ℓ 2, various examples. Be able to determine whether or not YTAX is a real inner product, given A. §0.2, §0.3
• Be able to compute the angle between two vectors, the length of a vector, and the distance between two vectors. §0.4.
2. Types of convergence (See Fourier series below.) Be able to define each type and to explain the differences between them.
• Pointwise convergence. §0.3.1
• Mean convergence. §0.3.1.
• Uniform convergence. §0.3.2.
• Orthogonality
1. Orthogonal and orthonormal sets of vectors, orthonormal bases, and orthogonal complements of subspaces. Know the definitions for these terms. Know how to write a vector in terms of an orthonormal basis, and how to calculate the coefficients. Be able to do problems similar to ones assigned in homework. §0.5.1
2. Orthogonal projections and least squares. Be able to find orthogonal projections and to solve minimization problems and least-squares fitting problems. §0.5.2, §0.7.1
3. Gram-Schmidt process. Be able to find an o.n. set from a given non-orthogonal set. §5.3

### Fourier Series

• 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 2π on an intervals of the form [−π, π], [0, 2π], or [c − π, c + π]. Be able to know and use Lemma 1.3.
3. 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].
4. Be able to use symmetry properties to help compute coefficients in FS, FSS, FSC.
• Convergence of Fourier series
1. Pointwise convergence
• Riemann-Lebesgue Lemma. Be able to give a proof of this in the simple case that f is continuously differentiable. §1.3.1.
• Know the conditions under which an FS, FSS, FCS are pointwise convergent. Be able to use them to decide what function an FS, FSS, or FCS converges to pointwise.
• Be able to briefly sketch the major steps in the proof of pointwise convergence, stating the roles played by the Fourier (Dirichlet) kernel and the Riemann-Lebesgue Lemma. (Algebraic details of derivations are not what I want here.)
• Be able to use the pointwise convergence to sum series. (See problem 21, p. 85 for an example.)
2. Uniform convergence
• Be able to define the term uniform convergence.
• Know the conditions under which an FS, FSS, or FCS is uniformly convergent, and be able to apply them.
• Be able to tell whether a series is only pointwise convergent or uniform convergent.
3. Mean convergence
• Projections and partial sums. (Lemma 1.3.4.)
• 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.
• Mean convergence theorem.

### Fourier Transforms

• Computing Fourier transforms & properties.
1. Be able to compute Fourier transforms and inverse Fourier transforms. Be able to use the properties in Theorem 2.6 to do this. §2.1-2.2.1
2. 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.)
3. Convolution theorem. Know the definition of the convolution and be able to do very simple examples. §2.2.2

Updated 2/19/2014.