MATH 322 Harmonic Analysis

Due Friday, March 23.

• Read Chapter 5, Section 1. Read Exercises #1--10. Do Exercises #1, 2, and 4 -- 10. Hand in #2, 6, 8, and 9.

Here are some comments on the exercises.

• Exercise 1. This is a nice exercise which brings together the theories of Fourier series and Fourier transform.
• Exercise 2. I did part of this in class. Comparison of f and g illustrates that greater smoothness is transformed into faster decay.
• Exercise 3. More about smoothness and decay.
• Exercise 4. I did this in class.
• Exercise 5. I did some of this in class. For (b), think about using the Riemann-Lebesque Lemma.
• Exercise 6. The functions you are asked to find are the eigenfunctions of the Fourier transform. As a hint, try multiplying the Gaussian by a polynomial. Look at Problem #7. (That's NOT Exercise #7.)
• Exercise 7. I did a similar computation in class.
• Exercise 8. The hint should be enough to get you started, but there is more to be done.
• Exercise 9. A direct analog to Fejer's Theorem for Fourier series.
• Exercise10. I guess this will be your third proof of the Weierstrass Theorem.