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
- 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.
- 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.
- Be able to use symmetry properties to help compute coefficients in
FS, FSS, FSC.
- Covergence of Fourier series
- Pointwise convergence
- Riemann-Lebesgue Lemma. Be able to give a proof of this in the
simple case that f is continuously differentiable. §1.3.1.
- 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.
- 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
- Know the conditions under which an FS, FSS, or FCS is uniformly
convergent, and be able to apply them.
- Gibbs' phenomenon. Be able to briefly describe the Gibbs'
phenomenon.
- Mean convergence
- Best approximation property of partial sums.
- 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.
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.)