## Math 641-600 Final Exam Review (Fall 2016)

The final exam will be given on Wednesday, Dec. 14, 8-10 am in our
usual classroom. It will cover sections 2.2.7, 3.2 - 3.6, 4.1 - 4.3.2,
and all class notes, starting from Splines and finite element spaces,
except for the notes on X-ray tomography. The test will consist of the
following: statements and/or proofs or sketches of proofs of theorems;
statements of definitions; proofs of short propositions or solutions
of problems similar to ones done in the
**
homework** or in class. Extra office hours: Friday (12/9), 2-4;
Monday (12/12), 1-3, Tuesday (12/13), 10-12 and 2-3. For other times,
send me an email to arrange an appointment.
### Finite elements and Fourier Series

- Fourier series (§2.2.3, Notes for 10/13)
- The finite element spaces S
^{h}(k,r) (§2.7,
Splines and Finite Element Spaces.)
- Cubic splines
- Interpolation
- Smoothing
- Differential equations

### Operators and integral equations

- Bounded operators (§3.2, Bounded
Operators & Closed Subspaces,
and
Projection theorem, the Riesz representation theorem, etc.)
- Norms of linear operators, unbounded operators, continuous linear
functionals, spaces associated with operators
- Hilbert-Schmidt kernels
- The Projection Theorem
- The Riesz Representation Theorem
- Existence of adjoints of bounded operators
- Fredholm alternative

- Compact operators (§3.3, §3.5, Compact
Operators and on
Closed Range Theorem.)
- Finite rank operators, $\mathcal C(\mathcal H)$ is a closed
subspace of $\mathcal B(\mathcal H)$ (Theorem 3.4), and
Hilbert-Schmidt kernels/operators
- Closed Range Theorem, Fredholm alternative, resolvents and
kernels

- Spectral theory for compact, self-adjoint operators, K = K*
(§3.4,
Spectral Theory for Compact Operators.)
- Eigenvalues and eigenspaces
- Eigenvalues are real; eigenvectors for distinct eigenvalues are
orthogonal
- Eigenspaces are finite dimensional
- The only limit point of the set of eigenvalues is 0.
- "Maximum principle" (p. 117)

- Completeness of eigenfunctions on the closure of the range of K
(Theorem 3.6)
- Application to eigenfunction problems involving integral equations

- Contraction Mapping Theorem, Neumann series (§3.6;)

### Distributions and differential operators

- Test function space D, distribution space D′, examples,
δ function, δ sequences, integral representation,
derivatives of distributions (§4.1, Example problems on distributions.)
- Green's functions for 2nd order operators (§4.2)
- Domain of an operator, adjoints of 2nd order
operators, domain of the adjoint (§4.3)

Updated 12/9/2016 (fjn).