Math 658-600 Midterm Review
The midterm will be given on October 17, 2012. The test will cover
sections 2.1-2.4, 2.6, 7.1-7.3, 7.5, 7.6. In addition, it will cover
material from my notes on both
the
Discrete Fourier Transform and
on
Filters. The test will have 3 or 4 questions, some with
multiple parts; questions will involve problems or propositions
similar
to
homework problems or those done in class. Be able to use the
basic theorems on Lebesgue integration discussed at the beginning of
the course.
Fourier series
- Be able to find the FS for various functions.
- Bessel's inequality. Riemann-Lebesgue lemma for f in
L2
- Pointwise convergence of FS for a PS, 2π periodic
function. Uniqueness of FS for a PS function
- Uniform convergence of FS. Derivatives, integrals of FS
- Gibbs' phenomenon, Cesàro means
- L2 theory of Fourier series
- Parseval/Plancheral theory
- Special spaces - C[a,b], Ck[a,b], L2,
ℓ2, L1
- Completeness of the set {ei n θ}, Parseval's
equation, convergence of FS in L2
Fourier transforms and convolutions
- Convolutions and approximate identities
- Special spaces - C[a,b], Ck[a,b], L2,
L1, L∞
- Properties of the FT and finding FTs - be able to find Fourier
transforms for various functions
- Convolution theorem, inversion theorem, Riemann-Lebesgue Lemma
- L2 theory, Parseval/Plancheral theorem
- Multivariate FT and radial functions - rotational
invariance, positive definite functions
- Applications
- Heat kernel/heat equation
- Sampling theorem
- Uncertainty principle
Filters
- Definition of time-invariant filter - cf. the notes on
Filters
- Causal filters - connection with the Laplace transform
- Examples
Discrete Fourier Transform
- DFT
and FFT
- Derivation from the trapezoidal rule
- Shifts, convolutions, and the convolution theorem
- FFT
Updated 10/15/2012 (fjn).