Math 414-501 — Spring 2015

Test 1 Review

General Information

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, 0.7.1, 1.2.1-1.2.5, and 1.3.4-1.3.5 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

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
     • Be able to compute the angle between two vectors, the length of a vector, and the distance between two vectors. §0.4.
• Orthogonality
  1. Orthogonal and orthonormal sets of vectors, orthonormal bases, and orthogonal complements, V^⊥. Know the definitions for these terms. Know how to write a vector in terms of an orthonormal basis, and how to calculate the coefficients. §0.5.1.
  2. Orthogonal projections and "least-squares" minimization problems. Be able to find orthogonal projections and to solve least-squares minimization problems. §0.5.2, §0.7.1.
  3. Gram-Schmidt process. Be able to use this to find a an orthonormal basis for a space. §0.5.3.

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. Other intervals. Fourier series for intervals of the form $[-a,a]$.
• Convergence of Fourier Series
  1. Pointwise convergence. S_N(x) → f(x) for each fixed x. Know the definitions of piecewise continuous and piecewise smooth functions. Be able to sketch the limit of $S_N(x)$ for a piecewise smooth, $2\pi$-periodic function and be able to evaluate series. Be able to use them to decide what function an FS, FSS, or FCS converges to. See class notes for 2/11/15, 2/13/15 and 2/16/15. §1.2.4
  2. Convergence in L² (mean convergence). ||f− S_N||_L² → 0.   § 0.5.2, §1.3.5
     • Be able to show that $S_N$ is the projection of $f$ onto the space $V_N = \text{span}\{1, \cos(x),\sin(x),\ldots,\cos(Nx),\sin(Nx)\}$, in the space $L^2[-\pi,\pi]$ (or $L^2[0,2\pi]$).
     • 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. (Theorem 1.35, Notes for 2/16/15.)
  3. Uniform convergence. S_N(x) → f(x) uniformly in x.
     • Be able to define precisely the term uniform convergence -- i.e., be able to define the phrase "converges uniformly in x." Be able to explain the difference between pointwise and uniform convergence. §1.3.4, Notes for 2/16/.15
     • 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.

Updated 2/17/2015.