MA5238 Fourier Analysis

1. Lecture 3. Tuesday 19 Jan 2010 MA5238 Fourier Analysis Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml/ http://arxiv.org/find/math/1/au:+Lawton_W/0/1/0/all/0/1

2. continuous and periodic with period Fourier Series The coefficients in this expansion are given by Question When is real-valued ?

3. continuous and periodic with period Fourier Series The coefficients in this expansion are given by and they satisfy Parseval’s identity

4. is continuous but NOT periodic choose define Fourier Integral Then

5. is absolutely integrable, we take the limit Fourier Integral Note error on page 30 in textbook

6. is sufficiently nice  we can define Fourier transform Fourier Transforms inverse Fourier transform

7. It follows from the preceding results that Fourier Transforms and and

8. Definition A function is rapidly decreasing if it satisfies any of the following equivalent conditions: Rapidly Decreasing Functions For any integer there exists such that For any polynomial Question Why are both of these conditions equivalent to the condition given on page 31 in the text ?

9. Definition The Schwartz class consists of all functions such that for every multi-index the function is rapidly decreasing. The Schwartz Class Question Which of the these functions are in Question Is

10. The Schwartz class (any n) is a vector space, Properties of is an algebra, is closed under multiplication by polynomials, is closed under differentiation, is closed under translations, is closed under multiplication by consists of integrable functions, extra: is closed under rotations and dilations.

11. Fourier transform Fourier Transform on inverse Fourier transform

12. Definition The convolution of functions is the function Convolution on Show that convolution satisfies these properties:

13. Function Fourier transform Properties of Fourier Transform on

14. Clearly hence is fast decreasing since for any polynomial Closed Under Fourier Transform Since is fast decreasing,

15. It suffices to prove Fourier Inversion On Question Why is the integration order reversible ? Integrating y first gives Integrating where first gives

16. It suffices to prove that This follows from the fact, proved later, that Fourier Inversion On since this family of functions is an approximate identity: 1. Note error on p.40 in text 2. 2.

17. Fourier Transform of Gaussian For above consider the contour As R increases, side_terms 0 and thus proving the previous formula for

18. Read Chapter 3 Do Problems 1-21 on pages 43-45 and prepare to solve on the board in class for Friday Assignment 3