Math 414501 — Spring 2014
Test 1 Review
General Information
Test 1 will be given on Monday, 2/24/2014. Please bring an
8½×11 bluebook. I will have office hours on
Thursday, 111 (Other times will be OK, but if I'm busy I'll tell you
to come back later.), Friday, 24, and on Monday 910.

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, 1.2, 1.3,
2.1, 2.2.1 and 2.2.2 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 and Fourier transform properties 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. Be able to determine whether or
not Y^{T}AX is a real inner product, given A. §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.
 Types of convergence (See Fourier series below.) Be able
to define each type and to explain the differences between them.
 Pointwise convergence. §0.3.1
 Mean convergence. §0.3.1.
 Uniform convergence. §0.3.2.
 Orthogonality
 Orthogonal and orthonormal sets of vectors, orthonormal bases,
and orthogonal complements of subspaces. Know the definitions for
these terms. Know how to write a vector in terms of an orthonormal
basis, and how to calculate the coefficients. Be able to do problems
similar to ones assigned in homework. §0.5.1
 Orthogonal projections and least squares. Be able to find
orthogonal projections and to solve minimization
problems and leastsquares fitting problems. §0.5.2, §0.7.1
 GramSchmidt process. Be able to find an o.n. set from a given
nonorthogonal set. §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].
 Be able to use symmetry properties to help compute coefficients
in FS, FSS, FSC.
 Convergence of Fourier series
 Pointwise convergence
 RiemannLebesgue Lemma. Be able to give a proof of this in the
simple case that f is continuously differentiable. §1.3.1.
 Know the conditions under which an FS, FSS, FCS are pointwise
convergent. Be able to use them to decide
what function an FS, FSS, or FCS converges to pointwise.
 Be able to briefly sketch the major steps in the proof of
pointwise convergence, stating the roles played by the Fourier
(Dirichlet) kernel and the RiemannLebesgue Lemma. (Algebraic details
of derivations are not what I want here.)
 Be able to use the pointwise convergence to sum series. (See
problem 21, p. 85 for an example.)
 Uniform convergence
 Be able to define the term uniform convergence.
 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.
 Mean convergence
 Projections and partial sums. (Lemma 1.3.4.)
 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.
Fourier Transforms
 Computing Fourier transforms & properties.
 Be able to compute Fourier transforms and inverse
Fourier transforms. Be able to use the properties in Theorem 2.6 to do
this. §2.12.2.1
 Be able to establish the simple properties listed in Theorem 2.6
of the text, and know how to use them. (You will be given a table
listing these properties plus a few others, so you do not need to
memorize them.)
 Convolution theorem. Know the definition of the convolution and
be able to do very simple examples. §2.2.2
Updated 2/19/2014.