Math 414-501 — Spring 2024

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, 1.2.1-1.2.5, and 1.3.1-1.3.5 in the text, along with the notes on pointwise convergence of FS. 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, 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.
     • 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, V^⊥. Know how to write a vector in terms of an orthonormal basis, and how to calculate the coefficients.
  2. Orthogonal projections and "least-squares" minimization problems. Be able to find orthogonal projections and to solve least-squares minimization problems.
  3. Gram-Schmidt process. Be able to use this to find an orthonormal basis for a space.

Fourier Series

• Calculating Fourier Series
  1. Extensions of functions — periodic, even periodic, and odd periodic extensions. Be able to sketch extensions of functions.
  2. Be able to compute Fourier series, in either real or complex forms, Fourier sine series and Fourier cosine series. Be able to use and prove Lemma 1.3.
  3. Other intervals. Fourier series for intervals of the form $[-a,a]$ and $[0,2a]$.
• Convergence of Fourier Series
  1. Pointwise convergence. Be able to define pointwise convergence for an FS, FSS, or FCS. 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 for a continuous, piecewise smooth $2\pi$-periodic function. Be able to use them to decide what function an FS, FSS, or FCS converges to. (You don't need the series itself to sketch the limit.)
     • Be able to to list its principal parts of the proof to briefly explain how the fit together. See the proof of pointwise convergence in the notes on pointwise convergence of Fourier series.
     • Riemann-Lebesgue Lemma. Be able to give a proof of this in the simple case that $f$ is continuously differentiable.

  2. Uniform convergence. Be able to define the term uniform convergence of an FS, FSS, FCS. Know the conditions under which one of these is uniformly convergent, and be able to apply them. Be able to tell whether a series is only pointwise convergent or uniform convergent. Be able to explain the difference between pointwise, uniform and mean convergence.

  3. Convergence in the mean (i.e., in L²). $\lim_{N\to \infty}$||f− S_N||_L² → 0.
     • Parseval's equation. Know both the real and complex form. Be able to use it to sum series similar to ones given in the homework or example done in class. Be able to show that, if $f\in L^2$, $S_N$ converges in the mean to $f$ if and only if Parseval's equation holds.
     • Mean convergence theorem. (Theorem 1.35.) If $f\in L^2$, then $S_N$ converges in the mean to $f$.

Updated 2/25/2024.