## 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 $\lambda, \lambda'\in \rho(L)$,
$R_{\lambda}(L)-R_{\lambda'}(L) = (\lambda-\lambda')R_\lambda(L)R_{\lambda'}(L).$
• 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.)
1. Definition and notation, derivatives of distributions, multiplication of a distribution by C functions increasing polynomially
2. Know the continuity 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

Updated 4/27/2018 (fjn).