Math 414501 — Final Exam Review
General Information
The final exam will be given on Friday, 5/8/2015, from 10:30 am to 12:30 pm. Please bring an
8½×11 bluebook. Extra office hours: TBA.

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 and coverage. There will be 5 to 7
questions, some with multiple parts. The test will cover the topics
listed below. Points on the test will be approximately distributed
this way: chapter 1, 10%; chapters 2 and 3, 20% and, chapters 46,
70%. Questions from chapters 13 will be involve calculations, but no
theory. Any theory from the remaining chapters will only be
derivations. In addition, there will be a table
of integrals and Fourier transform properties. Here are links to
practice tests:
2001 and 2009
Topics Covered
Fourier series. Be able to calculate the Fourier series for a
$2\pi$ periodic function, using either the real form or the complex
form, and to sketch the function to which the series
converges. Finally, be able to determine whether or not the
convergence is uniform.
Fourier transforms & filters. Be able to find Fourier
transforms, inverse Fourier transforms, and convolutions. Be able to
filter a simple signal. A table of integrals and Fourier transforms
will be supplied. §2.3.
Discrete Fourier transform. Be able to define the DFT and to
explain how the FFT is related to the DFT. Know how fast
(i.e., number of multiplications) the FFT is vs. the DFT.
Discretetime signals Be able to filter discretetime
signals using the discretetime convolution. (This really will only be
used in conjunction with MRA filters.)
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). Be able to define these quantities,
along with the corresponding wavelets, for the Haar and Shannon MRA's.
 $p_k$'s in Scaling relation. Know the properties of the
$p_k$'s listed in Theorem 5.9.
 Decomposition &
reconstruction. Know the decomposition and reconstruction
formulas, highpass and lowpass decomposition and reconstruction
filters, and how to down sample and up sample a signal.
 Know how to implement both decomposition and reconstruction
algorithms. In particular, be able to show that the top level
coefficients, which are used in the initialization step, have the
approximate form a^{j}_{k} ≈ m f( 2^{
j}k), where m = ∫_{∞}^{∞}
φ(x)dx (Theorem 5.12).

Fourier transform criteria for an MRA. Be
able to find the Fourier transformed form of the scaling function
and the wavelet. Be able to outline how the scaling function and the
wavelet are derived from the function P(z) that satisfies the
conditions §5.3.3, Theorem 5.1.9, & notes for 5/4/15.

Daubechies' wavelets. Know how the
Daubechies wavelets are classified using N, where 2N is the length
of the four filters and how how N relates to the number of vanishing
moments of a wavelet. For N=2, be able to explain how the
approximate form of the wavelet coefficients (equation (6.13)) can
be used in singularity detection and data compression. Be able to
explain various ways of handling the problem of overspill; see
section 6.3.
Updated 5/4/2015.