Math 641-600 Final Exam Review (Fall 2007)
The in-class part of the final will be given on December 7, 12:30-2:30
pm in our usual classroom. You have already received
the
take-home part of the exam. The test will cover sections 2.2.3-2.2.7,
3.2-3.6, and also the notes on
the discrete
Fourier transform. It will consist of the following: statements of
theorems and definitions; short problems or propositions similar
to homework
problems or examples done in class; and 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 MRA
- Finite elements, spline spaces Sh(k,r), and B-splines, Nm(x)
- Interpolation problems
- Boundary value problems
Operators and integral equations
- Bounded operators
- The Projection (Decomposition) Theorem - see class notes for 11/15/07.
- The Riesz Representation Theorem (Theorem 3.1), application to existence of weak solutions to boundary value problems
- Existence of adjoints of operators (Theorem 3.2), norms of operators, continuous linear functionals
- Compact operators
- Finite rank operators, Approximation Theorem (Theorem 3.4), and Hilbert-Schmidt operators
- Spectral theory for compact operators
- Completeness of eigenfunctions for compact operators (Theorem 3.6), application to eigenfunction problems involving differential equation
- Resolvents and kernels, Fredholm alternative/closed range theorem (Theorem 3.7)
- Contraction Mapping Theorem (Theorem 3.8), Neumann series, Galerkin methods, solving integral equations
Updated 12/3/07 (fjn).