Math 414 - Test I Review

General Information

Test I (Friday, March 5) will have 4 to 6 questions, some with multiple parts. Please bring an 8½×11 bluebook. Problems will be similar to ones done for homework. 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 either calculus or linear algebra. you will be given a table of integrals along with a list of the properties of the Fourier transform. I will announce extra office hours on my web site.

Topics Covered

Fourier Series

Calculating Fourier Series
1. Fourier series (FS). Given a function, you should be able to compute a Fourier series in either real or complex form, and with prescribed period 2a. If a ≠ π I will call your attention to the fact.
2. Fourier sine series (FSS) and Fourier cosine series (FCS). Be able to compute FSS and FCS for functions defined on a half interval, [0,a]. As above, if a ≠ π I will call your attention to the fact.
3. Be able to use symmetry properties to help compute coefficients in FS, FSS, FSC.
Covergence of Fourier series
- Pointwise convergence
  1. Riemann-Lebesgue Lemma. Be able to give a proof of this in the simple case that f is continuously differentiable. §1.3.1.
  2. 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.
  3. 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.)
- Uniform convergence
  1. Know the conditions under which an FS, FSS, or FCS is uniformly convergent, and be able to apply them.
  2. Gibbs' phenomenon. Be able to briefly describe the Gibbs' phenomenon.
- Mean convergence
  1. Best approximation property of partial sums.
  2. 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.
  3. Mean convergence theorem.

Fourier Fransforms

Computing Fourier transforms & properties. (§2.1-2.2.1) Be able to compute Fourier transforms and inverse Fourier transforms. 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.)