Math 414501 — Spring 2016
Test 1 Review
General Information
 Time and date. Test 1 will be given on Wednesday, 2/17/16,
at 10:20, in our usual classroom.
 Bluebooks. Please bring an 8½×11
bluebook.
 Office hours. I will have office hours on Monday
(2/15/16), 1:404, and on Tuesday (2/16/15), 11:301:30.

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 4 to 6 questions, some
with multiple parts. The test will cover sections 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
 We will only deal with 2π periodic functions.
 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 2π on an intervals of
the form [−π, π], [0, 2π], or [c − π, c +
π].
 Fourier sine series (FS for odd, 2πperiodic extension) and
Fourier cosine series (FS for even, 2πperiodic extension). Be
able to compute FSS and FCS for functions defined on a half interval,
[0,π].
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. §1.3.1.
 Fourier (Dirichlet) kernel, P. Know what P is and how to express
partial sums in terms of P. §1.3.2
 Be able to sketch a proof for pointwise convergence of a
FS, making use of the formula for P and the properties of P as well
as the RiemannLesbegue Lemma. §1.3.11.3.2. (To sketch a proof
means to list its principle parts. We did this in class: see the
notes for 2/1/162/5/16.)
 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. See
Exercise 21(b).
Uniform convergence
 Definition of uniform convergence.
 Conditions under which an FS, FSS, or FCS is uniformly
convergent. Be able to apply these to determine whether or not an FS
is uniformly convergent. These are all stated for periodic
functions. To apply them on [−π,π], or [0,π], work with the
appropriate periodic extension. §1.3.4.
 Gibbs' phenomenon. Be able to briefly describe the Gibbs'
phenomenon.
Mean convergence
 Definition of mean convergence.
 Mean approximation property of partial sums. (See notes for
2/12/16). Be able to show that if f is in L^{2}, with
complex Fourier coefficients α_{n}'s, then
\[
\int_{\pi}^\pi f(x)S_N(x)^2dx = \int_{\pi}^\pi f(x)^2dx 
2\pi \sum_{n=N}^N \alpha_n^2.
\]
 Parseval's theorem. Know both the real and complex form. Be able
to use this theorem to sum series. See Example 1.4.1.
 Here are the main theorems on mean convergence.
 If f is in L^{2}, then the partial sums of the FS for f
converge in the mean to f.
 If f is in L^{2} and Parseval's equation holds, then the
partial sums of the FS for f converge in the mean to f.
Updated 2/14/2016.