Math 414-501 Test 2 Review
General Information
Test 2 will be given on Wednesday, 4/7/10. Please bring an
8½×11 bluebook. I will have extra office hours
on Monday, 4/5/10, and Tuesday, 4/6/10 (TBA). Material covered
includes 1.3.4, 1.3.5, 2.1-2.4, and 3.1, except for sections 3.1.5 and
3.1.6.
-
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. There will be 4 to 6 questions, some with multiple
parts. The problems will be similar to ones done for homework, and as
examples in class and in the text. Both a brief table of integrals and
a list of properties of the Fourier transform will be supplied.
Topics Covered
Fourier Series
Uniform convergence
- Definitions. Be able to define these terms: uniform
convergence, piecewise smooth function, Gibbs' phenomenon.
- Conditions under which an FS, FSS, or FCS is uniformly
convergent. There are two types of conditions: Those that apply
to the series coefficients (Lemma 1.33) and those that apply to the
function itself (Theorem 1.30). Be able to apply these to determine
whether or not an FS is uniformly convergent. §1.3.4.
- Error estimates. Be able to do problems similar to those
for problem 3, Assignment 5.
Mean convergence
- Definitions. Be able to define these: mean
(L2) convergence, best approximation in L2.
- Theorems.
- Be able to state this. If f is in L2, then the partial
sums of the FS for f converge in the mean to f.
- Be able to prove Lemma 1.34, which gives a distance (or least
squares) minimizing property for partial sums of a Fourier series.
- Parseval's equation (theorem). Be able to state both the real and
complex form, and be able to use it to sum series similar to ones
given in the homework.
Fourier Transforms
- Definitions. Be able to define these: Fourier transform,
inverse Fourier transform, convolution, shift operator
Ta, linear time-invariant filter, causal filter, ideal
filter, response function, system function, Nyquist frequency,
Nyquist rate.
- Theorems.
- Be able to state and prove (or sketch a proof, as required) of
the following theorems: the convolution theorem for the Fourier
transform, the Sampling Theorem.
- Be able to state and use Plancherel's Theorem (Theorem 2.12), and
the conditions under which a filter is time invariant (Theorem 2.17),
or time invariant and causal (Theorem 2.19).
- The table of properties given in Theorem 2.6 will be supplied. Be
able to prove the properties that we didi in class.
- Calculations. Be able to find Fourier transforms;
inverse Fourier transforms; convolutions; outputs of filters - given
system response function and signal; integrals via Plancherel's
Theorem. You may use any property of the Fourier transform to do the
calculation. A brief table of integrals and the properties of the
Fourier transform in Theorem 2.6 will be supplied. The problems will
be similar to those done in class or for homework.
Discrete Fourier Analysis
- Definitions. Be able to define these: discrete Fourier
transform (DFT); FFT algorithm; discrete periodic
signals Sn; convolution of discrete periodic
signals.
- Theorems.
- Be able to state and prove (or sketch a proof, as required) of
the following theorems: convolution theorems for the DFT; the DFT
maps Sn; to itself (Lemma 3.2); the inversion
theorem for the DFT (sketch); the approximation of the FT via FFT
(eqn. (3.6), p. 144).
- Be able to state these; the number of operations needed to
calculate the FFT vs. matrix multiplication; the DFT in matrix form
(eqn. (3.2) and preceeding equation on p. 148).
Updated 4/3/2010.