Math 642 Final Exam Review Spring 2018
The final exam for Math 642 will be held on Friday, May 4, from
10:30-12:30, in our usual classroom. The test covers the material
from sections 7.1, 7.2, 10.2, 10.3, and also material on the spectral
theorem, Schwartz space and tempered distributions. Material from 6.5
will not be on the test. I'll have my usual office hours, and
additional ones that I will announce. There will be 5 to 7
questions. You will be asked to state a few definitions, and to do
problems similar to those assigned
for
homework, and examples done in class. In addition, you will be
asked to give or sketch a proof for a major theorem or lemma from the
material above.
Operators on Hilbert space
- Unbounded operators
- Be able to define the terms and be able to show the results
below.
- Operators. Densely defined, closed & closable, extensions,
adjoint, self adjoint, and resolvent operator $R_\lambda(L):=(L -
\lambda I)^{-1}$.
- Be able to prove that if $L$ is densely
defined, then $L^\ast$ is closed. Know that if $L$ is both closed and
densely defined, then so is $L^\ast$.
- Resolvent set and spectrum for a closed, densely defined operator
$L$
- Resolvent set $\rho(L)=\{\lambda\in \mathbb C: R_\lambda \in
\mathcal B(\mathcal H)\}$ or $R(L-\lambda I)=\mathcal H$.
- Spectrum $\sigma(L)=\rho^\complement(L)$. The discrete spectrum
$\sigma_d$ consists of all eigenvalues of $L$. The residual
spectrum $\sigma_r$ consists of all $\lambda$ such that $L-\lambda
I$ is one-to-one and $\overline{R(L-\lambda I)} \ne
\mathcal H$. The continuous spectrum $\sigma_c$ is composed of all
$\lambda$ such that $L-\lambda I$ is one-to-one and
$\overline{R(L-\lambda I)} = \mathcal H$. Finally,
$\sigma=\sigma_d\cup \sigma_r\cup \sigma_c$, where $\sigma_d$,
$\sigma_r$ and $\sigma_c$ are disjoint.
- Resolvent operators
- Be able to prove that $R_\lambda(L)$ is analytic, bounded
on the resolvent set.
- First Resolvent Identity: Be able to show this: If $L$ is
a closed, densely defined operator on a Hilbert space $\mathcal H$,
then, for $\lambda, \lambda'\in \rho(L)$,
$ R_{\lambda}(L)-R_{\lambda'}(L) =
(\lambda-\lambda')R_\lambda(L)R_{\lambda'}(L). $
- Self adjoint operators.
- Be able to show that if $L=L^\ast$, then $\sigma_r=
\emptyset$, and both $\sigma_d$ and $\sigma_c$ are subsets of $\mathbb
R$.
- Spectral theorem for $L=L^\ast$
- Be able to define the term spectral family,
Eλ. Be able to state the spectral theorem.
- Stone's formula, Green's functions, and spectral transform. Be
able to find the spectral transform in simple cases Fourier
transform, Fourier sine and cosine transforms.
Fourier transforms
- Definition of transform and inverse transform. (Use whichever
sign convention you want, just be consistent.)
- Be able to establish simple properties (Theorem 7.2).
- Be able to compute transforms and inverses of transforms, using
contour integration if necessary.
Know the convolution theorem and be able to establish simple
L1 properties of convolutions e.g., f,g ∈
L1 implies that f∗g ∈
L1.
- Be able to prove or sketch a proof for each of these:
- The convolution theorem
- The Plancheral/Parseval theorem
- ``Useful'' form of Plancheral/Parseval's Theorem.
$\int_{\mathbb R} f(u)\hat g(u)du = \int_{\mathbb R}
\hat f(u)g(u)du$.
- The Shannon Sampling Theorem
Schwartz space and tempered distributions
- Schwartz space $\mathcal S$.
(See Feldman's
online notes).
- Definition and notation
- Semi-norm and (equivalent) metric space topologies
- Know that the Fourier transform is a bijective map from $\mathcal
S$ into itself.
- Tempered distributions, $\mathcal S'$.
(See
Feldman's online notes.)
- Definition and notation, derivatives of distributions,
multiplication of a distribution by C∞ functions
increasing polynomially
- Know the continuity test for a linear functional to be a tempered
distribution (Feldman, Theorem 6, pg. 7)
- The Fourier transform of a tempered distribution is defined via
Parseval's identity,
$\int_{\mathbb R} T(u)\hat \phi(u)du = \int_{\mathbb R}
\hat T(u)\phi(u)du$
- Examples of Fourier transforms of tempered distributions
Updated 4/27/2018 (fjn).