## Math 642 Final Exam Review – Spring 2015

The final exam for Math 642 will be held on Monday, May 11, from 8 to 10, in our usual classroom (ARCC 205). The test covers the material from sections 7.1, 7.2, 10.1-10.3, as well as the material on tempered distributions that we discussed in class. I'll have office hours on Friday: 1-3, 4-5.

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, Eλ. 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.

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
• 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.)
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.
• 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).