## Math 642 Final Exam Review – Spring 2017

The final exam for Math 642 will be held on Monday, May 8, 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. 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
assigned homework problems (starting with assignment 7) 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
- Terms: densely defined, closed & closable, extensions,
adjoint, self adjoint, resolvent operator, resolvent set, spectrum
(discrete, continuous, and residual)

- Resolvent operator (L − λI)
^{−1}
- Analytic, bounded on the resolvent set.
- Spectrum of a self-adjoint operator — real; no residual spectrum.

- Spectral theorem
- Be able to define the term spectral family,
E
_{λ}. 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.
Know the convolution theorem and be able to establish simple
L
^{1} properties of convolutions — e.g., f,g ∈
L^{1} implies that f∗g ∈
L^{1}.
- 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.
$\int_{\mathbb R} f(u)\hat g(u)du = \int_{\mathbb R}
\hat f(u)g(u)du$.

**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,
multiplcation of a distribution by C
^{∞} functions
increasing polynomially
- Know the coninuity 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

**Asymptotic analysis**

- Asymptotic estimates and series; big "O" and little "o" notation
- Watson's lemma. Be able to state, prove, and use it.
- Suggested problems. See
assignment 9.

Updated 5/3/2017 (fjn).