Math 414-501 Spring 2021
Test 3 Review
General Information
- Time and date. Test 3 will be given on Wednesday, 4/23/21,
at 1:35, in our usual classroom.
- Bluebooks. Please bring an 8½×11
bluebook.
- Office hours. I will have office hours on Wednesday
(4/21) from 12 to 1 and on Thursday (4/22/21) from 3 to 4.
-
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 calculus, or of 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 all of chapter 4 and sections
5.1, 5.2, 5.3.3 and 5.3.4. The problems will be similar to ones done
for
homework especially assignments 8 and 9 , and as
examples in class and in the text. Here are links to practice
tests:
2001, 2009
and 2016
Be aware that these tests cover some material that will not be on the
test for this class. The formulas that will be provided on the test
may be found at this
link: test
3 formulas
Topics Covered
Haar Wavelet Analysis
- Haar scaling function and approximation spaces. Know
what the Haar scaling function, $\phi$, is and be able to derive its
two-scale relation. Be able to define its corresponding approximation
spaces $V_j$. Know the nesting and scaling properties for these
spaces. Be able to use the $\{\phi(2^jx-k\}_{k=-\infty}^\infty$ basis
for $V_j$. §§4.2.1-4.2.2
- Haar wavelet and wavelet spaces. (§4.2.4) Know the
definition of the Haar wavelet and Haar wavelet spaces Wj,
along with their properties. §4.2.3
- Decomposition and reconstruction. Be able to do simple
decomposition and reconstruction problems similar to the ones done
for homework. §4.3.1, §4.3.2
Multiresolution Analysis (MRA)
- Mallat's MRA. Be define Mallat's multiresolution analysis,
including the approximation spaces (V's), the scaling relation (in
terms of $p_k$'s), the wavelet spaces (W's), and the wavelet itself
(again, in terms of $p_k$'s).
- Haar & Shannon MRA's. Be able to describe in detail both
the Haar and Shannon MRA's. For each, state what the approximation
spaces, wavelet spaces, scaling function and wavelets are. Be able to
verify that the properties of an MRA are satisfied for both
cases. (In the Shannon case, able to state and use the sampling
theorem.)
- $\mathbf {p_k}$'s in Scaling relation. Be able to derive simple
properties of the the scaling function, including that
$\{2^{j/2}\phi(2^j-k)\}_{k=-\infty}^\infty$ is an o.n. basis for
$V_j$. Be able to derive the scaling relation. Be able to briefly
explain how to get wavelet relation.
- Decomposition & reconstruction. Be able to derive the
decomposition formulas and to use them to obtain the high pass and low
pass filters in the decomposition filter diagram. Be able to use
reconstruction filter diagrams in simple cases.
- Fourier transform criteria for an MRA. Be able to
find the Fourier transformed form of the scaling function and the
wavelet given in Theorem 5.19 and equation 5.30, and outline how the
scaling function and the wavelet are derived from the function P(z)
that satisfies the conditions §5.3.3, Theorem 5.23.
Updated 4/20/2021