Math 641-600 Final Exam Review (Fall 2016)

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).