Math 642 Final Exam Review Spring 2006
The final exam for Math 642 will be held on Monday, May 8, from 3 to 5 pm in our usual classroom, MILN 313. The test covers these sections from the text: 7.3, 8.2, and 10.1-10.5. In addition, it will cover the material on tempered distributions and Schwartz space that I lectured on. This is outlined below.
Schwartz space and tempered distributions
- Schwartz space, S
- Definition
- 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 4 in assignment 5.) 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 distribution is defined via Parseval's identity,
∫R T(u)f^(u)du = ∫R T^(u)f(u)du
- Weak temperate convergence
- The Fourier transform takes weakly temperate convergent sequences into weakly temperate convergent sequences.
- Differentiation takes weakly temperate convergent sequences into weakly temperate convergent sequences.
- Theorem. The Fourier transform of S′ is S′.
- Convolutions of distributions, the convolution theorem, and convolution-type Fredholm equations
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 5/4/06 (fjn).