Math 414-501 — Spring 2022

Test 2 Review

General Information

Time and date. Test 2 will be given on Wednesday, 3/23/22, at 11:30, in our usual classroom.

Bluebooks. Please bring an 8½×11 bluebook.

Office hours. I will have office hours on Monday (3/21/22), 12:30-1:1:30 & 2-3, and on Tuesday (3/22/22), 11:30-1:30 & 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 calculus, or of 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 0.2-0.5, 1.3.4-1.3.5, and 2.1. The problems will be similar to ones done for homework, and as examples and proofs done in class and in the text. A short table of integrals will be provided. Here are links to practice tests: 2009 and 2010. Be aware that these tests cover some material that will not be on the test for this class.

Topics Covered

Fourier Series

• Uniform convergence
  1. Be able to define the term uniform convergence.
  2. Know the conditions given in Theorem 1.30 under which an FS (or FCS or FSS) is uniformly convergent. Be able to apply these to determine whether or not an FS is uniformly convergent.

• Mean convergence
  1. Be able to state the definition of mean convergence.
  2. Parseval's equation. Know both the real and complex form, Theorems 1.39 and 1.40. Be able to use this theorem to sum series.

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
     • Given an inner product, be able to compute the angle between two vectors, the length of a vector, and the distance between two vectors.
• 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.
  2. Orthogonal projections and least squares. Know the definition of an orthogonal projection and be able to find them. Be able to show that $S_N$, the partial sum of a Fourier series for a periodic function $f$, is the projection of $f$ onto the span of $\{1/\sqrt{2\pi}, \cos(x)/\sqrt{2\pi}, \sin(x)/\sqrt{2\pi},\cdots,\cos(Nx)/\sqrt{2\pi},\sin(Nx)/\sqrt{2\pi}\}$.
  3. Gram-Schmidt process. Be able to find an o.n. set from a given non-orthogonal set.

Fourier Transforms

• Computing Fourier transforms & properties
  1. Be able to compute Fourier transforms and inverse Fourier transforms.

Updated 3/20/2022.