Math 335 Fourier Theory Notes

Short Description
Fourier theory is a branch of mathematics first invented to solve certain problems in. partial differential equations. The most well-known of these …

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Content
Macquarie University
Department of Mathematics
Fourr iierr Theorry
B..M..N.. CllarrkeTable of Contents
1. Introduction …………………………………………………………………………………… ……………………………1
2. Linear differential operators …………………………………………………………………………………… ……3
3. Separation of variables …………………………………………………………………………………… ……………5
4. Fourier Series …………………………………………………………………………………… …………………………9
5. Bessel’s inequality …………………………………………………………………………………… …………………14
6. Convergence results for Fourier series …………………………………………………………………………16
7. Differentiation and Integration of Fourier Series ………………………………………………………….20
8. Half-range Fourier series …………………………………………………………………………………… ………23
9. General Intervals …………………………………………………………………………………… …………………25
10. Application to Laplace’s equation ………………………………………………………………………………..27
11. Sturm-Liouville problems and orthogonal functions ……………………………………………………..33
12. Bessel Functions …………………………………………………………………………………… ………………….. 37
13. Fourier Transforms …………………………………………………………………………………… ………………41
14. Inverse Fourier Transforms …………………………………………………………………………………… …..47
15. Applications to Differential Equations …………………………………………………………………………50
16. Plancherel’s and Parseval’s Identities ………………………………………………………………………….54
17. Band Limited Functions and Shannon’s Sampling Theorem ………………………………………….56
18. Heisenberg’s Inequality …………………………………………………………………………………… …………59…