Math 641-600 Final Exam Review
The in-class part of the final will be given on Wednesday, December
16, 10:30-12:30 in our usual classroom. The final will also be given
on December 9, from 7-9 pm, in Milner Hall. The test will cover
sections 2.2.3-2.2.7, 3.2-3.4, 4.1-4.3.2, 4.5 (pp. 161-163) in the
text, my notes on
the discrete
Fourier transform and the
Daubechies wavelets, and any material covered in class. The test
will be composed of two parts. The first part will consist of
statements of theorems and definitions; the second will have short
problems or propositions similar
to homework
problems or examples done in class, as well as a proof of one of
the major theorems highlighted in red
below.
Approximation tools
- Fourier series
- Discrete
Fourier transform
- Sampling theorem, sinc functions
- Wavelets – MRA, scaling function & wavelet, two-scale
relation, Haar
wavelets,
Daubechies wavelets
- Finite elements, spline spaces Sh(k,r), and B-splines,
Nm(x), finite element method for boundary value problems
Operators and integral equations
- Bounded operators
- Norms of linear operators, unbounded operators, continuous linear
functionals, spaces associated with operators
- Integral equations -- Hilbert-Schmidt kernels, Fredholm, Volterra
- Projection Theorem (class notes,
11/18/09), projections onto a subspace
- The Riesz Representation Theorem
- Adjoints of operators
- Weak form of a boundary value problem
- Compact operators
- Finite rank operators, Approximation
Theorem (Theorem 3.4), and Hilbert-Schmidt operators,
- Closed range theorem (Theorem
3.7), Fredholm alternative
- Spectral theory for compact operators
- Eigenvalues, eigenspaces
- Completeness of eigenfunctions
(Theorem 3.6)
- Application to eigenfunction problems involving integral equations
Distributions and applications
- Test function space D, distribution space D′, examples,
δ function, integral representation, derivatives of
distributions
- Green's functions
Updated 12/7/09 (fjn).