**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. - Kodaira/Weyl/Stone 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
- Uncertainty principle

**Schwartz space and tempered distributions**

- Schwartz space
*S*(See Feldman's online notes).- Definition and notation
- Semi-norm and (equivalent) metric space topologies
- Theorem. The Fourier transform of
*S*is*S*. Be able to sketch a proof of this.

- Tempered distributions,
*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.
- Laplace's method. Be able to derive asymptotic estimates, e.g. Stirling's formula.
- Practice problems. See assignment 9.

** Structure of the exam**

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 6) 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.

Updated 5/8/2015 (fjn).