Math 642 Final Exam Review Spring 2010
The final exam for Math 642 will be held on Monday, May 10, from 10:30 to 112:30 am in our usual classroom, ZACH 119D. The test covers these sections from the text: 6.5 (Gamma function, Bessel functions, generating functions), 7.1 (spectral theory) 7.2.1 (Fourier transforms and main properties), 10.1-10.3 (Laplace's method, Watson's Lemma, and examples). In addition, it will cover the material on self-adjoint operators, Schwartz space, and tempered distributions that I lectured on. This material is outlined below.
Spectral theory for self-adjoint operators
- Definition self-adjoint operator
- Spectrum of a self-adjoint operator real; no residual spectrum
- Spectral theorem
- Kodaira/Weyl/Stone formula, Green's functions, and spectral transform
Schwartz space and tempered distributions
- Schwartz space S
- Definition and notation
- Semi-norm and (equivalent) metric space topologies
- Theorem. S is dense in L2(R).
- Theorem. The Fourier transform of S is S.
- Various useful results: A polynomial × a Schwartz function is a Schwartz function. One may also multiply by certain C∞ functions and still have a Schwartz function. (See problem 1 in assignment 7.) Translates of Schwartz functions are Schwartz functions. Convolutions of Schwartz functions are Schwartz functions.
- ``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.)
- Tempered distributions, S′
- 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
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 covered by this test.
Updated 5/4/2010 (fjn).