Math 641-600 Final Exam Review
The in-class part of the final will be given on December 12, 8-10 am in our usual classroom. You have already received the take-home part of the exam. The test will cover sections 2.2.4-2.2.7, 3.2-3.6, 4.1-4.3.2, 4.5 (pp. 161-163). It will also cover the notes on the the discrete Fourier transform. The in-class 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
- Discrete Fourier transform
- Fourier transform, sampling theorem, sinc functions
- Wavelets, MRA
- Finite elements, spline spaces Sh(k,r), and B-splines, Nm(x)
- Weak formulation of boundary value problems
Operators and integral equations
- Types of integral equations -- Fredholm, Volterra
- Bounded operators
- The Projection (Decomposition) Theorem
- The Riesz Representation Theorem
- Adjoints of operators, norms of operators, continuous linear functionals
- Compact operators
- Finite rank operators, Approximation Theorem (Theorem 3.4), and Hilbert-Schmidt operators
- Spectral theory
- Resolvents and kernels, Fredholm alternative (Theorem 3.7)
- Contraction Mapping Theorem, Neumann series, Galerkin methods, solving integral equations
Differential operators
- Distributions
- Green's functions
- Unbounded operators, domain of an operator, adjoints
- Eigenfunction expansions
Updated 12/6/06 (fjn).