Math 414-501 Spring 2021
Test 1 Review
General Information
- Time and date. Test 1 will be given on Friday, 2/26/21,
at 1:35, in our usual classroom.
- Bluebooks. Please bring an 8½×11
bluebook.
-
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 1.2.1-1.3.5 in the
text,
the
sine and cosine series, and
the notes
on point-wise convergence of 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. 1/25/21, 1/27/21
- 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. §1.2.3, the notes
on
sine and cosine series 1/27/21
Pointwise convergence
- Definition of pointwise convergence.
- Definitions of piecewise continuous, jump discontinuity, and
piecewise smooth.
- Proof of pointwise convegence. The material below, which was
covered in the classes on 2/5, 2/8 and 2/10, comes from
the notes
on point-wise convergence of Fourier series.
- Riemann-Lebesgue Lemma. Be able to give a proof of this in the
simple case that $f$ is continuously differentiable.
- Fourier (Dirichlet) kernel, $P_N$. Know how to express
partial sums in terms of $P_N$.
- 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 Riemann-Lesbegue Lemma. (To sketch a proof means to
list its principle parts, but without details of their proofs.)
- Be able to use the theorems on pointwise convergence (Theorem
1.22 & 1.28) to decide the function that an FS, FSS, or FCS
converges to.
- Be able to use pointwise convergence to sum a series. See the
examples from class and in the notes mentioned above. 1/27/21
Uniform convergence
- Definition of uniform convergence.
- Be able to prove Theorem 1.30, under the condition that $f$ is
twice continuously differentiable.
- Know the 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. §1.3.4, 2/12/21.
- Gibbs' phenomenon. Be able to briefly describe the Gibbs'
phenomenon. Problem 1.32, 2/12/21
Mean convergence
- Be able to state the definition of mean convergence.
- Parseval's equation. Know both the real and complex form,
Theorems 1.39 and 1.40. Be able to use this theorem to sum series. See
Example 1.4.1.
Updated 2/21/2021.