Math 641-600 (Fall 2014) — Final Exam Review
The final will be given from 8 am to 10 am on Tuesday, December 16,
2014, in our usual classroom. The test will cover sections 2.2.3,
2.2.7, 3.2-3.6, 4.1, 4.2 4.3.1-4.3.2 (2nd order operators, only), 4.5
(pp. 161-163) 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
- Fourier series, point-wise convergence, Parseval's theorem (skip
DFT)
- 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
- D, convergence in D, "bump" function, D′, convergence in
D′, δ function, derivatives of distributions
- Green's functions
- Second order differential operators: domain, adjoint (including
domain), self-adjoint operators
- Completeness of Sturm-Liouville eigenfunctions; see notes for
12/8/14.
Updated 12/9/14 (fjn).