Math 414-501 Spring 2017
Test 1 Review
- Time and date. Test 1 will be given on Wednesday, 2/15/17,
at 10:20, in our usual classroom.
- Bluebooks. Please bring an 8½×11
- Office hours. I will have office hours on Monday
(2/13/17), 11:10-1:30, and on Tuesday (2/14/17), 11:30-1:30.
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 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 problems will be similar to ones done
homework, and as examples in class and in the text. A
table of integrals will be provided. Here are links to practice
Be aware that these tests cover some material that will not be on
the test for this class.
Calculating Fourier Series
- Extensions of functions periodic, even periodic, and odd
periodic extensions. Be able to sketch extensions of
- 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
- Definition of pointwise convergence.
- Definitions of piecewise continuous, jump discontinuity, and
- Riemann-Lebesgue Lemma. Be able to give a proof of this in the
simple case that f is continuously differentiable. §1.3.1. (I
want the proof given on 2/3/17.)
- 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 Riemann-Lesbegue Lemma. §1.3.1-1.3.2. (To sketch a proof
means to list its principle parts. We did this in class: see the
notes for 2/1/17-2/3/17.)
- Be able to use the theorems on pointwise convergence (Theorem
1.22 & 1.28) to decide what function an FS, FSS, or FCS
- Be able to use pointwise convergence to sum a series.
- 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
- Be able to state definition of mean convergence.
- 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 L2, then the partial sums of the FS for f
converge in the mean to f.
- If f is in L2 and Parseval's equation holds, then the
partial sums of the FS for f converge in the mean to f.