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

  1. 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
  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
  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)

  1. 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).
  2. 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.)
  3. $\mathbf {p_k}$'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.
  4. 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.
  5. 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