Distributions and tempered distributions

• Spaces of test functions
  • The space $\mathscr D(\mathbb R^n)$
    • Definition, multi-index notation, examples (§9.1)
    • Convergence in $\mathscr D$. (Notes for 10/24/12 and part (ii) of the definition on p. 306 in the text).
    • Operations mapping $\mathscr D$ continuously to itself — differentiation, multiplication by a $C^\infty$ function, convolutions, and so on. (§9.2)
  • Schwartz space $\mathscr S(\mathbb R^n)$
    • Definition and notation (§9.4)
    • Semi-norm and (equivalent) metric space topologies (See HW 6 and notes, 10/26/12)
    • Operations continuously mapping $\mathscr S$ to itself — differentiation, multiplication by polynomials, Fourier transform, and so on.
    • Be able to sketch a proof of this theorem: The Fourier transform on $\mathscr S$ is a bijection continuous in the topology of $\mathscr S$. (See text and notes, 10/29/12.)
• The spaces $\mathscr D'$ and $\mathscr S'$
  • Distributions, $\mathscr D'$
    • Definition and notation, examples, $\delta$ function, and convergence of a sequence of distributions
    • Operations on $\mathscr D'$ — differentiation multiplication by a $C^\infty$ function, and so on. (§9.1 - §9.3)
  • Tempered distributions, $\mathscr S'$
    • Know the definition as a continuous linear functional on $\mathscr S$ (see notes, 10/31/12 and 11/2/12), and also know notation examples, and convergence of sequences of tempered distributions (§9.3 and HW 7., 8.)
    • Know the definition of a finite order tempered distribution and the know and be able to use continuity test: A linear functional defined on $\mathscr S$ is in $F\in \mathscr S'$ if and only if $F$ is of finite order. (Notes, 11/2/12.)
    • Operations on $\mathscr S'$ — differentiation, multiplication by polynomials, taking Fourier transforms, etc.
    • Be able to find Fourier transforms of tempered distributions. (See HW 6 and HW 7.)

Reproducing kernel Hilbert spaces (RKHS)

• Definition, simple properties, and examples (notes, 11/7/12 and 11/9/12)
  • Be able to the existence of the associated reproducing kernel
  • Interpolation and connection with least-squares minimization problems (notes, 11/7/12 and 11/9/12)
• Positive definite and conditionally positive definite kernels
  • Associated RKHS and inner product (notes, 11/12/12)

Scattered-data interpolation via RBFs

• Scattered data interpolation problems. (See the Notes on Scattered-Data Radial Function Interpolation. Theorems, propositions, etc. all refer to these notes.))
  • Interpolation without polynomial reproduction
  • Interpolation with polynomial reproduction
  • Polynomials and unisolvent sets
• Strictly positive definite and conditionally positive definite functions (see Definition 2.1)
  • Be able to prove Proposition 2.2.
  • Completely monotonic functions (Widder's theorem)
  • Radial functions that are positive definite for all $\mathbb R^n$'s. Schoenberg's theorem — be able use it to give examples
  • Order $m$ RBFs — Theorem 2.5. See examples in §III.
  • Native spaces and the power function (notes, 11/19/12 and HW 8)

Spherical harmonics

• Laplace-Beltrami for $\mathbb S^2$. (See the Notes on Spherical Harmonics.)
  • Spherical coordinates and how to handle singularities at the poles.
• The eigenvalue problem
  • Be able to get the bound in Theorem 2.1
  • Be able to sketch a proof of Proposition 2.2
• The spherical harmonics. (You do not need to memorize the formulas in in sections 2 and 3.)
  • Know how to use raising and lowering operators to go from one spherical harmonic to another
  • Know that when homogeneous harmonic polynomials on $\mathbb R^3$ are restricted to the sphere $\mathbb S^2$ they are spherical harmonics.

Rotations in $\mathbb R^3$

• Definition and properties of 3×3 rotation matrices
  • Be able to show that the eigenvalues of a rotation matrix $R$ satisfy $|\lambda|=1$ and that $\lambda =1 $ is always an eigenvalue
  • SKIP the Euler angles
  • Be able to give a geometric derivation of the formula $R\mathbf x = \mathbf x + \sin(\theta)\mathbf \omega \times \mathbf x+ (1-\cos(\theta))\mathbf \omega\times (\mathbf \omega \times \mathbf x)$. (Notes, 12/3/12.)
• SU(2) and SO(3)
  • Be able to express $U\in SU(2)$ in terms of quaternions and to use it to obtain the connection between $SU(2)$ and the sphere $\mathbb S^3$.
  • Be able to do a problem similar to problem 7 in HW 9. (You won't be asked to do any long calculations.)
  • Know the connection between $SU(2)$ and $SO(3)$ and the one between $SO(3)$ and the projective space $\mathbb RP^3$.

Updated 12/7/2012 (fjn).