Math 414 - Test I Review
General Information
Test I (Thursday, March 6) will have 5 to 7 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.
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 2π.
- Fourier sine series (FSS) and Fourier cosine series (FCS). Be
able to compute FSS and FCS for functions defined on a half interval,
[0,π].
- 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.
- 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 transforms
- Computing Fourier transforms & properties.
(§2.1-2.2) Be able to compute Fourier transforms and inverse
Fourier transforms. Be able to establish the simple properties listed
on pg. 100 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.) Be able to the convolution theorem.
- Linear filters. (§2.3) Know what a linear,
time-invariant filter is, and what its system response function and
system function are. Be able to define the term causal
filter.
- The Sampling Theorem. (§2.4) Be able to state and
prove this theorem.
Discrete Fourier analysis
- Discrete Fourier transform. (§3.1) Be able to
define the DFT and inverse DFT, and be able to explain how it can be
used to approximate coefficients in a Fourier series. Be able to
use this to derive the DFT approxiamtion to the Fourier transform of a
function.
- FFT. (&3.1.3-3.1.4) Know what the connection between the DFT
and FFT is, and be able to explain why the FFT is "fast".