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Course11110451960 lectures

Introduction to Fourier Analysis

Prof. Parasar Mohanty · IIT Kanpur

Start with lecture 1 At NPTEL

About

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

Week 1

  1. Lecture 01: Origin of Fourier series YouTube
  2. Lecture 02: Convergence of a series and Riemann integration YouTube
  3. Lecture 03: Riemann integration and periodic functions YouTube
  4. Lecture 04: Fourier coefficients and series YouTube
  5. Lecture 05: Complex Fourier series YouTube

Week 2

  1. Lecture 06: Riemann Lebesgue lemma and Dirichlet kernel YouTube
  2. Lecture 07: Convolution of two Riemann integrable functions YouTube
  3. Lecture 08: Properties of convolution YouTube
  4. Lecture 09: Cesaro summability and summation by parts YouTube
  5. Lecture 10: Fejer kernel YouTube

Week 3

  1. Lecture 11: Fejer theorem and applications YouTube
  2. Lecture 12: Good kernels and Poisson kernel YouTube
  3. Lecture 13: Abel summability YouTube
  4. Lecture 14: Dirichlet problem YouTube
  5. Lecture 15: Convergence at jump discontinuity YouTube

Week 4

  1. Lecture 16: Orthonormal families YouTube
  2. Lecture 17: Pythagoras theorem and Parseval identity YouTube
  3. Lecture 18: Pointwise convergence YouTube
  4. Lecture 19: Parseval identity applications YouTube
  5. Lecture 20: Divergent Fourier series YouTube

Week 5

  1. Lecture 21: Pointwise convergence of S_N(f) YouTube
  2. Lecture 22: Isoperimetric problem YouTube
  3. Lecture 23: Weyl's equidistribution theorem YouTube
  4. Lecture 24: on Equidistributed sequences YouTube
  5. Lecture 25: Proof of Weyl's criterion YouTube

Week 6

  1. Lecture 26: Fourier analysis on finite groups YouTube
  2. Lecture 27: Fourier transform on Z(N) YouTube
  3. Lecture 28: Inversion theorem and Parseval identity YouTube
  4. Lecture 29: Results on Fourier coefficients of two functions YouTube
  5. Lecture 30: Fast Fourier transform YouTube

Week 7

  1. Lecture 31: Fourier analysis on finite abelian group YouTube
  2. Lecture 32: Fourier analysis on finite abelian groups YouTube
  3. Lecture 33: Simultaneously diagonalizable operators and characters YouTube
  4. Lecture 34: Results of Fourier series on finite abelian group YouTube
  5. Lecture 35: Applications in Number theory YouTube

Week 8

  1. Lecture 36: Fourier transform on R YouTube
  2. Lecture 37: Properties of Fourier transform YouTube
  3. Lecture 38: Inversion formula and convolution YouTube
  4. Lecture 39: Fejer kernel on R YouTube
  5. Lecture 40: Schwartz space YouTube

Week 9

  1. Lecture 41: Convolution and Good kernel in Schwartz space YouTube
  2. Lecture 42: Multiplication formula and Fourier inversion theorem YouTube
  3. Lecture 43: Plancherel formula and Poisson summation formula YouTube
  4. Lecture 44: Application of Poisson summation formula YouTube
  5. Lecture 45: Weierstrass theorem and Heisenberg uncertainty principle YouTube

Week 10

  1. Lecture 46: Hermite operator YouTube
  2. Lecture 47: Solution of ODE by using Fourier transform YouTube
  3. Lecture 48: Laplacian equation YouTube
  4. Lecture 49: Poisson kermal and mean value theorem for harmonic functions YouTube
  5. Lecture 50: Wave equation YouTube

Week 11

  1. Lecture 51: Eigenvalues of Fourier transform YouTube
  2. Lecture 52: Fourier transform on R^n YouTube
  3. Lecture 53: Properties of Fourier transform on R^n YouTube
  4. Lecture 54: Inversion theorem and Plancherel theorem YouTube
  5. Lecture 55: Wave equation, heat equation and Poisson kernel YouTube

Week 12

  1. Lecture 56: Fourier series in higher dimension YouTube
  2. Lecture 57: Poission summation formula YouTube
  3. Lecture 58: Application of Poission summation formula YouTube
  4. Lecture 59: Radon transform YouTube
  5. Lecture 60: Reconstruction formula YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Kanpur
Department
Mathematics
Teacher
Prof. Parasar Mohanty
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge.
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