Math 414501 — Spring 2018
Test 1 Review
General Information
 Time and date. Test 1 will be given on Monday, 2/19/18,
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 (2/15/18), 12:30, on Friday
(2/16/18), 2:303:30.

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 sections 0.5.1 and
1.2.11.3.5 in the text. 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 three periods of
periodic extensions of functions.
 Fourier series. Be able to compute Fourier series in either real
or complex forms, and with prescribed period $2\pi$ on an intervals of
the form $[\pi,\pi]$, $[0, 2\pi]$. You may be asked to use other
intervals, $[a,a]$ for some value of $a$. (I'll make it clear to you
if $a\ne \pi$.)
 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.
 RiemannLebesgue Lemma. Be able to give a proof of this in the
simple case that f is continuously differentiable. (We did the more
general piecewise smooth case in class on 2/2/18.) §1.3.1.
 Fourier (Dirichlet) kernel, $P_N$. Know what $P_N$ is and be able
to express $S_N$ in terms of $P_N$. §1.3.2
 Be able to sketch a proof for pointwise convergence of a
FS, making use of the formula for $P_N$ and the properties of $P_N$
as well as the RiemannLesbegue Lemma. §1.3.11.3.2. (To sketch
a proof means to list its principle parts.)
 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.
 Be able to use pointwise convergence to sum a series.
Uniform convergence
 Definition of uniform convergence.
 Conditions under which an FS, FSS, or FCS is uniformly convergent
— Theorem 1.30 (on the function) Lemma 1.33 (on the
coefficients), and Be able to apply these to determine whether or not
an FS is uniformly convergent. These are all stated for periodic
functions. §1.3.4.
Mean convergence
 Know the definitions of an orthogonal set of vectors, an
orthonormal set of vectors, orthogonal projection. Be able to use
Theorem 0.21 to find the orthogonal projection of a vector onto a
subspace. (See Assignment 3, problem 4.)
 Be able to state definition of mean convergence.
 Mean convergence.. Let $f\in
L^2[\pi,\pi]$ (or $L^2[0,2\pi]$). Then the partial sums $S_N$ for $f$
converge in the mean to $f$, and Parseval's Equation holds.
 Parseval's theorem. Know both the real (Eqn. 1.40) and complex
form (Eqn. 1.41). Be able to use this theorem to sum series. See
Example 1.4.1.
Updated 2/15/2018.