# Introduction to Fourier Analysis

- Institution: IIT Kanpur
- Department: Mathematics
- Teacher: Prof. Parasar Mohanty
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111104519)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111104519)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

This course will cover the fundamental theory of Fourier series and Fourier integrals, as well as Fourier series for finite abelian groups. Some applications of Fourier series in differential equations, the isoperimetric problem, and number theory will be explored. Basic calculus and Riemann integrals will be required as prerequisites. The book “Introduction to Fourier Analysis” by E. Stein and R. Shakarchi will serve as the basis for this course.

Intended audience: Undergraduate

Prerequisites: Basic Calculus, Riemann Integration

## Lectures

1. [Lecture 01: Origin of Fourier series](https://invisiblenetwork.ai/learn/nptel/111104519/9ac3c9f716c2) (YouTube)
2. [Lecture 02: Convergence of a series and Riemann integration](https://invisiblenetwork.ai/learn/nptel/111104519/581420639c89) (YouTube)
3. [Lecture 03: Riemann integration and periodic functions](https://invisiblenetwork.ai/learn/nptel/111104519/726d4abab3df) (YouTube)
4. [Lecture 04: Fourier coefficients and series](https://invisiblenetwork.ai/learn/nptel/111104519/3cdf9d7e205a) (YouTube)
5. [Lecture 05: Complex Fourier series](https://invisiblenetwork.ai/learn/nptel/111104519/0b9374cbeed6) (YouTube)
6. [Lecture 06: Riemann Lebesgue lemma and Dirichlet kernel](https://invisiblenetwork.ai/learn/nptel/111104519/68b5e0a172b1) (YouTube)
7. [Lecture 07: Convolution of two Riemann integrable functions](https://invisiblenetwork.ai/learn/nptel/111104519/3ae51086dd0c) (YouTube)
8. [Lecture 08: Properties of convolution](https://invisiblenetwork.ai/learn/nptel/111104519/3c8d38fb624b) (YouTube)
9. [Lecture 09: Cesaro summability and summation by parts](https://invisiblenetwork.ai/learn/nptel/111104519/52e6f21e5a7c) (YouTube)
10. [Lecture 10: Fejer kernel](https://invisiblenetwork.ai/learn/nptel/111104519/ea393850c391) (YouTube)
11. [Lecture 11: Fejer theorem and applications](https://invisiblenetwork.ai/learn/nptel/111104519/91aa811ad9fa) (YouTube)
12. [Lecture 12: Good kernels and Poisson kernel](https://invisiblenetwork.ai/learn/nptel/111104519/6c13dfc48dc6) (YouTube)
13. [Lecture 13: Abel summability](https://invisiblenetwork.ai/learn/nptel/111104519/0bb9c57f95e8) (YouTube)
14. [Lecture 14: Dirichlet problem](https://invisiblenetwork.ai/learn/nptel/111104519/f7ba8f83525e) (YouTube)
15. [Lecture 15: Convergence at jump discontinuity](https://invisiblenetwork.ai/learn/nptel/111104519/bf092470baf4) (YouTube)
16. [Lecture 16: Orthonormal families](https://invisiblenetwork.ai/learn/nptel/111104519/2f341bf8c138) (YouTube)
17. [Lecture 17: Pythagoras theorem and Parseval identity](https://invisiblenetwork.ai/learn/nptel/111104519/03d5ccb89ac7) (YouTube)
18. [Lecture 18: Pointwise convergence](https://invisiblenetwork.ai/learn/nptel/111104519/c5801134e39b) (YouTube)
19. [Lecture 19: Parseval identity applications](https://invisiblenetwork.ai/learn/nptel/111104519/a844cf3d0bb7) (YouTube)
20. [Lecture 20: Divergent Fourier series](https://invisiblenetwork.ai/learn/nptel/111104519/1941004b5a5c) (YouTube)
21. [Lecture 21: Pointwise convergence of S_N(f)](https://invisiblenetwork.ai/learn/nptel/111104519/234ef44525bd) (YouTube)
22. [Lecture 22: Isoperimetric problem](https://invisiblenetwork.ai/learn/nptel/111104519/e60bee606cd0) (YouTube)
23. [Lecture 23: Weyl's equidistribution theorem](https://invisiblenetwork.ai/learn/nptel/111104519/a018e78ba3ea) (YouTube)
24. [Lecture 24: on Equidistributed sequences](https://invisiblenetwork.ai/learn/nptel/111104519/d26be174b5dc) (YouTube)
25. [Lecture 25: Proof of Weyl's criterion](https://invisiblenetwork.ai/learn/nptel/111104519/70786c8fa8f2) (YouTube)
26. [Lecture 26: Fourier analysis on finite groups](https://invisiblenetwork.ai/learn/nptel/111104519/2d90c7899288) (YouTube)
27. [Lecture 27: Fourier transform on Z(N)](https://invisiblenetwork.ai/learn/nptel/111104519/4a3dac8b8093) (YouTube)
28. [Lecture 28: Inversion theorem and Parseval identity](https://invisiblenetwork.ai/learn/nptel/111104519/49c667490c4e) (YouTube)
29. [Lecture 29: Results on Fourier coefficients of two functions](https://invisiblenetwork.ai/learn/nptel/111104519/91f981c3b149) (YouTube)
30. [Lecture 30: Fast Fourier transform](https://invisiblenetwork.ai/learn/nptel/111104519/0c00e776cd94) (YouTube)
31. [Lecture 31: Fourier analysis on finite abelian group](https://invisiblenetwork.ai/learn/nptel/111104519/3b934e504a9e) (YouTube)
32. [Lecture 32: Fourier analysis on finite abelian groups](https://invisiblenetwork.ai/learn/nptel/111104519/e22abaa1a4c6) (YouTube)
33. [Lecture 33: Simultaneously diagonalizable operators and characters](https://invisiblenetwork.ai/learn/nptel/111104519/9e237ad9e3c5) (YouTube)
34. [Lecture 34: Results of Fourier series on finite abelian group](https://invisiblenetwork.ai/learn/nptel/111104519/466d01fc7e7f) (YouTube)
35. [Lecture 35: Applications in Number theory](https://invisiblenetwork.ai/learn/nptel/111104519/6acd1ae318b8) (YouTube)
36. [Lecture 36: Fourier transform on R](https://invisiblenetwork.ai/learn/nptel/111104519/6b1948c52fb4) (YouTube)
37. [Lecture 37: Properties of Fourier transform](https://invisiblenetwork.ai/learn/nptel/111104519/892fca15aafb) (YouTube)
38. [Lecture 38: Inversion formula and convolution](https://invisiblenetwork.ai/learn/nptel/111104519/f40aba4abc5b) (YouTube)
39. [Lecture 39: Fejer kernel on R](https://invisiblenetwork.ai/learn/nptel/111104519/ee50400da513) (YouTube)
40. [Lecture 40: Schwartz space](https://invisiblenetwork.ai/learn/nptel/111104519/b67589300727) (YouTube)
41. [Lecture 41: Convolution and Good kernel in Schwartz space](https://invisiblenetwork.ai/learn/nptel/111104519/f3ae3be6882e) (YouTube)
42. [Lecture 42: Multiplication formula and Fourier inversion theorem](https://invisiblenetwork.ai/learn/nptel/111104519/1eca20cddc3d) (YouTube)
43. [Lecture 43: Plancherel formula and Poisson summation formula](https://invisiblenetwork.ai/learn/nptel/111104519/93203d2d8ac2) (YouTube)
44. [Lecture 44: Application of Poisson summation formula](https://invisiblenetwork.ai/learn/nptel/111104519/4ed434838fb2) (YouTube)
45. [Lecture 45: Weierstrass theorem and Heisenberg uncertainty principle](https://invisiblenetwork.ai/learn/nptel/111104519/7aa959c72d54) (YouTube)
46. [Lecture 46: Hermite operator](https://invisiblenetwork.ai/learn/nptel/111104519/f67402a9a9c3) (YouTube)
47. [Lecture 47: Solution of ODE by using Fourier transform](https://invisiblenetwork.ai/learn/nptel/111104519/4c83ae29dafe) (YouTube)
48. [Lecture 48: Laplacian equation](https://invisiblenetwork.ai/learn/nptel/111104519/8f50095a1a03) (YouTube)
49. [Lecture 49: Poisson kermal and mean value theorem for harmonic functions](https://invisiblenetwork.ai/learn/nptel/111104519/706d3cffb117) (YouTube)
50. [Lecture 50: Wave equation](https://invisiblenetwork.ai/learn/nptel/111104519/716db6dd34db) (YouTube)
51. [Lecture 51: Eigenvalues of Fourier transform](https://invisiblenetwork.ai/learn/nptel/111104519/dcd3efbcc35a) (YouTube)
52. [Lecture 52: Fourier transform on R^n](https://invisiblenetwork.ai/learn/nptel/111104519/59ae469b6ea9) (YouTube)
53. [Lecture 53: Properties of Fourier transform on R^n](https://invisiblenetwork.ai/learn/nptel/111104519/78b1b0e2bb9e) (YouTube)
54. [Lecture 54: Inversion theorem and Plancherel theorem](https://invisiblenetwork.ai/learn/nptel/111104519/5f71209dfcfc) (YouTube)
55. [Lecture 55: Wave equation, heat equation and Poisson kernel](https://invisiblenetwork.ai/learn/nptel/111104519/ba6ec0aeb4e0) (YouTube)
56. [Lecture 56: Fourier series in higher dimension](https://invisiblenetwork.ai/learn/nptel/111104519/ce987a66685e) (YouTube)
57. [Lecture 57: Poission summation formula](https://invisiblenetwork.ai/learn/nptel/111104519/69326ae8e6a9) (YouTube)
58. [Lecture 58: Application of Poission summation formula](https://invisiblenetwork.ai/learn/nptel/111104519/5d3287e30422) (YouTube)
59. [Lecture 59: Radon transform](https://invisiblenetwork.ai/learn/nptel/111104519/4942d1287a4f) (YouTube)
60. [Lecture 60: Reconstruction formula](https://invisiblenetwork.ai/learn/nptel/111104519/f2f26d403258) (YouTube)