**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,
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.

- Be able to define the term spectral family,
E

**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 — f,g ∈ L^{1}implies that f∗g ∈ L^{1}. - 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.)
- 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

- Definition and notation, derivatives of distributions,
multiplcation of a distribution by C

**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).