Math 414501 — Spring 2015
Test 1 Review
General Information
Test 1 will be given on Friday, 2/20/15. Please bring an
8½×11 bluebook. I will have office hours on
Wednesday (2/18/15), 23:30, and on Thursday (2/19/15), 10:301:30.

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 0.10.5, 0.7.1,
1.2.11.2.5, and 1.3.41.3.5 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
Topics Covered
Inner Product Spaces
 Inner products
 Definitions of real and complex inner products, examples of inner
product spaces.
 Standard inner products
on R^{n}, C^{n}, L^{2} and
ℓ^{ 2}, various examples. §0.2, §0.3
 Be able to compute the angle between two vectors, the length of a
vector, and the distance between two vectors. §0.4.
 Orthogonality
 Orthogonal and orthonormal sets of vectors, orthonormal bases,
and orthogonal complements, V^{⊥}. Know the definitions
for these terms. Know how to write a vector in terms of an orthonormal
basis, and how to calculate the coefficients. §0.5.1.
 Orthogonal projections and "leastsquares" minimization
problems. Be able to find orthogonal projections and to solve
leastsquares minimization problems. §0.5.2, §0.7.1.
 GramSchmidt process. Be able to use this to find a an
orthonormal basis for a space. §0.5.3.
Fourier Series
 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 2π on an intervals of
the form [−π, π], [0, 2π], or [c − π, c +
π]. Be able to know and use Lemma 1.3.
 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].
 Other intervals. Fourier series for intervals of the form
$[a,a]$.
 Convergence of Fourier Series
 Pointwise convergence. S_{N}(x) → f(x) for
each fixed x. Know the definitions of piecewise continuous
and piecewise smooth functions. Be able to sketch the limit of
$S_N(x)$ for a piecewise smooth, $2\pi$periodic function and be
able to evaluate series. Be able to use them to decide what function
an FS, FSS, or FCS converges to. See class notes for 2/11/15, 2/13/15
and 2/16/15. §1.2.4
 Convergence in L^{2} (mean convergence). f−
S_{N}_{L2} → 0. § 0.5.2,
§1.3.5
 Be able to show that $S_N$ is the projection of $f$ onto the
space $V_N = \text{span}\{1,
\cos(x),\sin(x),\ldots,\cos(Nx),\sin(Nx)\}$,
in the space $L^2[\pi,\pi]$ (or $L^2[0,2\pi]$).
 Parseval's theorem. Know both the real and complex form. be able
to use it to sum series similar to ones given in the homework.
 Mean convergence theorem. (Theorem 1.35, Notes for 2/16/15.)
 Uniform convergence. S_{N}(x) →
f(x) uniformly in x.
 Be able to define precisely the term uniform convergence
 i.e., be able to define the phrase "converges uniformly
in x." Be able to explain the difference between pointwise and
uniform convergence. §1.3.4, Notes for 2/16/.15
 Know the conditions under which an FS, FSS, or FCS is uniformly
convergent, and be able to apply them. Be able to tell whether a
series is only pointwise convergent or uniform convergent.
Updated 2/17/2015.