1. MA5238 Fourier Analysis Wayne Lawton Department of Mathematics S17-08-17, 65162749 matwml@nus.edu.sgmatwml@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 Lecture 3. Tuesday 19 Jan 2010

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

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

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

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

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

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

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

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

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

11. Fourier Transform on Fourier transform inverse Fourier transform

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

13. Properties of Fourier Transform on Fourier transformFunction

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

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

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

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

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