Math 641-600 (Fall 2015) — Final Exam Review
You have a choice for the date of the final exam. You may either take
it at the scheduled time, Wednesday, December 16, 10:30 am-12:30 pm,
in our usual classroom, or on Thursday, December 10, 7:30 pm-9:30
pm. (The room will be announced later.)
The test will cover sections
2.2.4, 2.2.7, 3.2-3.6, 4.1, 4.2 in the
text, class notes, 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 blue below.
Approximation tools
- Discrete Fourier transform, convolution
theorem, FFT
- Finite elements, spline spaces S^{h}(k,r), linear
splines, cubic interpolating splines, Galerkin method (skip
B-splines)
Operators
- Bounded operators
- B(H), bounded operators on H, continuous linear
functionals
- Examples: finite-rank operators, Hilbert-Schmidt operators, norms
of Hilbert-Schmidt operators
- Closed subspaces: null spaces, orthogonal complements, etc.
- Projection Theorem
- The Riesz Representation Theorem
- Adjoints of operators
- Weak form of a boundary value problem
- Fredholm Alternative,
applications to solving integral equations
- Compact operators
- C(H) is closed
in B(H).
- Finite rank operators, and Hilbert-Schmidt operators
- Closed range theorem, Fredholm
alternative for L = I − λK, application to integral
equations
- Spectral theory for compact operators
- Eigenvalues, eigenspaces
- Completeness of eigenfunctions
- Resolvents and resolvent kernels, solutions via eigenfunction
expansions
- Contraction Mapping Theorem,
Neumann series
Distributions and Differential Operators
- Green's functions, eigenvalue problems, and completeness of
eigenfunctions (Section 4.2, Keener)
- D, convergence in D, "bump" function, D′, convergence in
D′, δ function, derivatives of distributions (Section 4.1,
Keener. This will include only what we cover on Wednesday, December
8.)
Updated 12/7/15 (fjn).