Midterm Review Sheet--Math 414-500
General Information. The test will have 5 to 7 questions, some
with multiple parts. It will cover Chapters 0-2 of the
text. Please bring an 8½x11 bluebook. 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
either calculus or linear algebra. You will be provided with a table
of integrals. The questions will involve knowing definitions of terms
(15 points), doing computations, derivations, or applications (65-70
points), and statements and/or proofs of theorems (15-20
points). Questions may combine some of these; e.g., "Define uniform
convergence, and determine whether the series below is uniformly
convergent."
Terms. Here is a list of terms that you should be able to
define.
- inner product space
- orthogonal and orthonormal sets, orthogonal projection
- linear transformation
- mean, pointwise, and uniform convergence
- complete orthonormal set of vectors (i.e., take one away, and it
no longer spans).
- least squares
- Gibbs' phenomenon
- Parseval/Plancheral equations for Fourier series and Fourier
transforms
- time-invariant transformation
- filter, causal filter
Computations, derivations, or applications. Given a function,
you should be able to compute a Fourier series in either real or
complex form, and with prescibed period that may be different from
2
. You should also know how to compute
Fourier sine series and Fourier cosine series, and you should
understand the relation of these series to Fourier series in
general. You should understand the concept of evenness and oddness,
and be able to make use of any symmetry when computing coefficients
for the various series mentioned above. Be able to sum a series using
Parseval's theorem or by making use of what you know about pointwise
convergence. Be able to find Fourier transforms, either directly via
the definition or by means of the properties (a list will be
supplied). Be able to derive simple properties of the Fourier
transform, to find convolutions of functions, use Plancheral's
theorem, and to do simple filtering problems. Be able to determine if
a sampling rate is sufficient to reproduce a function, or will result
in aliasing. Be able to estimate the frequency dispersion for signal,
given its dispersion in time. In general, be able to work problems
similar to ones assigned from §1.4 and §2.6.
Statements and/or proofs of theorems. Here is a list.
- Pointwise convergence of Fourier series. (Be able to sketch the
steps in the proof.)
- Schwarz's inequality and the triangle inequality.
- Sampling Theorem.
- Uncertainty Principle.