Math 658-600 Test 1 Review
Test 1 will be given on October 3. The test will cover sections
2.1-2.4, 2.6, and 3.1-3.3. It will also cover material I covered in
class, including that on analog and digital filters. The test will
consist problems or propositions similar
to
homework problems or those done in class.
Fourier series
- Be able to find the FS for various functions.
- Bessel's inequality. Riemann-Lebesgue lemma for f in
L2. Least-squares minimization properties. (Assignment 1,
prob. 2.)
- 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. Convolutions.
L2 theory of Fourier series
- Convergent sequence, Cauchy sequence, complete spaces - Hilbert
spaces and Banach spaces
- Lebesgue integrals, sets of measure 0
- 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 L^2.
Filters
- Definition of time-invariant filter (continuous or discrete time)
- cf. notes
on filters
- Analog filters
- Digital filters
- Examples
Updated 10/1/2008 (fjn).