Math 642 Final Exam Review Spring 2011
The final exam for Math 642 will be held on Monday, May 9, from 10:30 to 12:30, in our usual classroom (CE 222). The test covers the material outlined below. I'll have these extra office hours: Thursday, 9-11; Friday, 10-1; Monday, 9-10.
Operators on Hilbert space
- Unbounded operators (See notes, 3/21/11, 3/23/11.)
- Terms: densely defined, closed & closable, extensions, adjoint, self adjoint.
- Be able to show that L* is closed.
- Spectrum of an operator L, σ(L)
- Discrete, continuous, residual. Know that σ(L) is a closed set.
- Resolvent set of L, ρ(L). Know that ρ(L) is an open set.
- Resolvent operator (L − λI)−1
- 1st resolvent identity (notes, 3/28/11). Be able to state and use this.
- Analytic, bounded on the resolvent set.
- Spectrum of a self-adjoint operator real; no residual spectrum.
- Connection with projections (notes, 4/1/11).
- Spectral theorem
- Be able to define the term spectral family, Eλ. Be able to state the spectral theorem (notes, 4/4/11, 4/6/11)
- Kodaira/Weyl/Stone formula, Green's functions, and spectral transform. Be able to find the spectral transform in simple cases (notes, 4/8/11, 4/11/11).
Fourier transforms
- Definition of transform and inverse transform (notes, 4/13/11, 4/15/11; section 7.2.1). (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.
- ``Useful'' form of Parseval's Theorem. ∫R f(u)g^(u)du = ∫R f^(u)g(u)du (f^ and g^ are the Fourier transforms of f and g.)
- Uncertainty principle. Be able to prove this.
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 (notes, 4/22/11).
- Tempered distributions, S′ (See Feldman's online notes.)
- Definition and notation
- Derivatives multiples of distributions
- The Fourier transform of a tempered distribution is defined via Parseval's identity,
∫R T(u)f^(u)du = ∫R T^(u)f(u)du
- 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.
- Examples. See assignment 7.
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 5) 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 5/4/2011 (fjn).