## Math 642 Final Exam Review – Spring 2016

The final exam for Math 642 will be held on Friday, May 6, 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 graphs of operators, the spectral theorem, 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)
• Graphs and operators
• 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, 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. Know the convolution theorem and be able to establish simple L1 properties of convolutions — f,g ∈ L1 implies that f∗g ∈ L1.
• Be able to prove or sketch a proof for each of these:
• Useful'' form of Parseval's Theorem.   $\int_{\mathbb R} f(u)\hat g(u)du = \int_{\mathbb R} \hat f(u)g(u)du$.
• Sampling theorem (See Keener, 7.2.3; Keener proves the theorem, but never gives it a name. I'll prove it on 5/3/16))
Let $f$ be in $L^2(\mathbb R)$ and suppose that the support of $\hat f(\xi)$ is contained in an interval $[-\Omega,\Omega]$. Then, if $\nu =\Omega/\pi$, $f(t) = \sum_{k=-\infty}^\infty f(k/\nu) {\rm sinc}(\nu t - k),$ where ${\rm sinc}(x):=\frac{\sin(\pi x)}{\pi x}$, $x\ne 0$ and ${\rm sinc}(0):=1$. ($\nu$ is called the Nyquist rate.)

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.)
1. Definition and notation, derivatives of distributions, multiplcation of a distribution by C functions increasing polynomially
2. Know the coninuity test for a linear functional to be a tempered distribution (Feldman, Theorem 6, pg. 7)
3. The Fourier transform of a tempered distribution is defined via Parseval's identity,
$\int_{\mathbb R} T(u)\hat \phi(u)du = \int_{\mathbb R} \hat T(u)\phi(u)du$
4. 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 5/3/2016 (fjn).