Math 414-501 Spring 2019
Test 1 Review
General Information
- Time and date. Test 1 will be given on Monday, 2/18/19,
at 10:20, in our usual classroom.
- Bluebooks. Please bring an 8½×11
bluebook.
- Office hours. In addition to my usual office hours, I will
have office hours on Thursday (2/14/19), 3-4:30, Friday (2/15/19),
and on Monday (2/18/19), 9-9:50.
-
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 4 to 6 questions, some
with multiple parts. The test will cover sections 1.2.1-1.3.5 in the
text, my notes on
Fourier sine and cosine
series, pointwise
convergence of Fourier series,
and least
squares and Fourier series. The problems will be similar to ones
done
for
homework, and as examples in class and in the text. A
short
table of integrals will be provided. Here are links to practice
tests:
2002,
2003
and 2009.
Be aware that these tests cover some material that will not be on
the test for this class.
Topics Covered
Calculating Fourier Series
- Extensions of functions periodic, even periodic, and odd
periodic extensions. Be able to sketch extensions of functions.
- Fourier series. Be able to compute Fourier series in either real
or complex forms, and with prescribed period $2\pi$ on intervals of
the form $[-\pi,\pi]$ or $[0, 2\pi]$. (We will not use $a\ne\pi$.)
- Fourier sine series (FSS) and Fourier cosine series (FCS). Be
able to compute FSS and FCS for functions defined on a half interval.
Pointwise convergence
- Definition of pointwise convergence.
- Definitions of piecewise continuous, jump discontinuity, and
piecewise smooth.
- Riemann-Lebesgue Lemma. Be able to give a proof of this in the
simple case that f is continuously differentiable. Know the proof in
my notes
on
pointwise convergence of Fourier series.
- Fourier (Dirichlet) kernel, $P_N$. Know what $P_N$ is and how to express
partial sums in terms of $P_N$. §1.3.2 and my notes
on
pointwise convergence of Fourier series.
- Be able to sketch a proof for pointwise convergence of a
FS i.e., list its principal parts. (We did this in class: see
the notes for 2/4/19.)
- Be able to use the theorems on pointwise convergence (Theorem
1.22 & 1.28) to decide what function an FS, FSS, or FCS
converges to, and to evaluate series. (See Exercise 21(b).)
Uniform convergence
- Definition of uniform convergence.
- Conditions under which an FS, FSS, or FCS is uniformly
convergent. §1.3.4.
- Gibbs' phenomenon. Be able to briefly describe the Gibbs'
phenomenon.
Mean convergence
- Definition of mean convergence.
- Minimization property of partial sums. This done in one of two
ways. The first is given in my notes on
least squares and Fourier series. The second was done in class
on Monday, 2/11/19 (see Theorem 0.5.20). These are also done in
Assignment 4, problems 1 and 2. §1.3.5.
- Parseval's theorem. Know both the real and complex form. Be able
to use this theorem to evaluate a series. See Example 1.4.1 and
similar problems in the homework.
- Here are the main theorems on mean convergence. Be able to state
them.
- If f is in L2, then the partial
sums of the FS for f converge in the mean to f.
- If f is in L2 and Parseval's equation holds, then the
partial sums of the FS for f converge in the mean to f.
Updated 2/12/2019.