Math 414-501 — Spring 2020

Test 1 Review

General Information

Time and date. Test 1 will be given at 12:40 pm on Wednesday, 2/26/2020, in our usual classroom.

Bluebooks. Please bring an 8½×11 bluebook.

Office hours I will have office hours on Monday (2/24), 1:30-4 and on Tuesday (2/25), 12-2.

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 0.1-0.3.1, 0.4-0.5, 1.2, 1.3, my notes on Examples for inner products, Fourier sine and cosine series, and on Pointwise convergence of Fourier series. The problems will be similar to ones done for homework, and as examples in class and in the text. Here are links to practice tests: 2002, 2003, 2009, and 2015.

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. §0.2, §0.3.1
     • In any given inner product, be able to compute the angle between two vectors, the length of a vector, and the distance between two vectors. See Examples for inner products and problem 3 in Assignment 1
• 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
  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. Know the Orthogonality Relations and be able to use them to show that $S_N=\text{Proj}_{V_N}(f)$. See the class notes for 2/5/20, 2/19/20, and also see Lemma 1.3.4.
  2. Fourier series. Be able to compute Fourier series in either real or complex forms, and with prescribed period 2π on intervals of the form $[-\pi, \pi]$, $[0, 2\pi]$. Be able to know and use Lemma 1.3. Be able to use symmetry (even, odd functions) properties to help compute coefficients in a Fourier series.
  3. Extensions of functions — periodic, even periodic, and odd periodic extensions. Be able to sketch extensions of functions.
  4. Fourier sine series (FS for odd, $2\pi$-periodic extension) and Fourier cosine series (FS for even, $2\pi$-periodic extension). Be able to compute FSS and FCS for functions defined on a half interval, $[0,\pi]$. See the notes Fourier sine and cosine series.
• Convergence of Fourier series
  1. Pointwise convergence
     • Be able to prove/derive any of the the formulas in the steps for the pointwise convergence theorem given in Pointwise convergence of Fourier series. Specifically, be able to do the following: express $S_N$ in terms of $P_N$; derive the properties of $P_N$; prove the Riemann-Lebesgue Lemma in the simple case that $f$ is continuously differentiable, for $\sin(\lambda x)$, $e^{i\lambda x}$, as well as $\cos(\lambda x)$; use these things to prove pointwise covergence.
     • Be able to use the theorem to decide what function an FS, FSS, or FCS converges to pointwise.
     • Be able to use the pointwise convergence to sum series. (See the class notes for 2/12/2020, problem 21, p. 85, and my notes Pointwise convergence of Fourier series for examples.)
  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.
     • Gibbs' phenomenon. See problem 32, pg. 87
     • Be able to tell whether a series is only pointwise convergent or uniform convergent.
  3. Mean convergence
     • Definition of mean convergence and when it holds (Theorems 1.35 and 1.36). See §1.3.5
     • Parseval's theorem. Know it in both real and complex form, and be able to use it to sum series similar to ones given in the homework. See §1.3.5

Updated 2/22/2020.