## 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 Sh(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).