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Course11110116465 lectures

Fourier Analysis and its applications

Prof. G. K Srinivasan · IIT Bombay

Start with lecture 1 At NPTEL

Lectures

week-01

  1. Lecture 1: Genesis and a little history Video
  2. Lecture 2: Basic convergence theorem Video
  3. Lecture 3: Riemann Lebesgue Lemma Video
  4. Lecture 4: The ubiquitous Gaussian Video
  5. Lecture 5: Jacobi theta function identity. Video

week-02

  1. Lecture 6: The Riemann zeta function Video
  2. Lecture 7: Bessel's functions of the first kind Video
  3. Lecture 8: Least square approximation Video
  4. Lecture 9: Parseval formula. Isoperimetric theorem Video
  5. Lecture 10: Dirichlet problem for a disc Video

week-03

  1. Lecture 11: The Poisson kernel Video
  2. Lecture 12: Cesaro summability and Fejer's theorem Video
  3. Lecture 13: Fejer's theorem (Cond) Video
  4. Lecture 14: Kronecker's theorem Video
  5. Lecture 15: Weyl's equidistribution theorem Video

week-04

  1. Lecture 16: Borel's theorem and beyond Video
  2. Lecture 17: Fourier transform and Schwartz space Video
  3. Lecture 18: Hermite's differential equation Video
  4. Lecture 19: Fourier inversion theorem Video
  5. Lecture 20: Plancherel's theorem Video

week-05

  1. Lecture 21: Heat equation. The heat kernel Video
  2. Lecture 22: Airy's function Video
  3. Lecture 23: Exercises on Fourier transform Video
  4. Lecture 24: Principle of equipartitioning of energy Video
  5. Lecture 25: Formula of Srinivasa Ramanujan Video

week-06

  1. Lecture 26: Sturm Liouville problems. Orthogonal systems Video
  2. Lecture 27: Vibrations of a circular membrane Video
  3. Lecture 28: Fourier Bessel series Video
  4. Lecture 29: Properties of Legendre polynomials Video
  5. Lecture 30: Properties of Legendre polynomials (contd) Video

week-07

  1. Lecture 31: Properties of Legendre polynomials interlacing of zeros Video
  2. Lecture 32: Laplace's integrals and Rayleigh's expansion of plane waves Video
  3. Lecture 33: Regular SturmLiouville problems Video
  4. Lecture 34: Variational properties of eigenvalues Video
  5. Lecture 35: The Dirichlet principle Video

week-08

  1. Lecture 36: Regular SturmLiouville problems Existence of eigenvalues Video
  2. Lecture 37: The Bergman space Video
  3. Lecture 38: The BanachSteinhaus' theorem Video
  4. Lecture 39: Hilbert space basics Video
  5. Lecture 40: Completeness of Hermite functions Video

week-09

  1. Lecture 41 : Laugerre polynomials Video
  2. Lecture 42: Orthonormal bases in Hilbert spaces Video
  3. Lecture 43: Nonseparable Hilbertspaces. Almost periodic functions Video
  4. Lecture 44: HilbertSchmidt operators and Green's functions Video
  5. Lecture 45: Spectrum of a bounded linear operator Video

week-10

  1. Lecture 46: Weak (sequential) compactness of the closed unit ball Video
  2. Lecture 47: Compact selfadjoint operators. Existence of eigen values Video
  3. Lecture 48: Compact selfadjoint operators. Existence of eigen values continued Video
  4. Lecture 49: Celestial Mechanics Video
  5. Lecture 50: Inverting the Kepler equation using Fourier series Video

week-11

  1. Lecture 51: Odds and Ends Video
  2. Lecture 52: Dirichlet's theorem on Fouier series Video
  3. Lecture 53: Dirichlet's theorem on Fouier series continued Video
  4. Lecture 54: Topology on the Schwartz space Video
  5. Lecture 55: Examples of tempered distributions Video

week-12

  1. Lecture 56: Operations on distributions Video
  2. Lecture 57: Fourier Transform of tempered distribution Video
  3. Lecture 58: Support of a distribution. Distributions with point support Video
  4. Lecture 59: Distributional solutions of ODEs. Continuity of the Fourier transform and differentiation Video
  5. Lecture 60: The Poisson summation formula Video

Live Session

  1. Live Session 13-03-2024 Video
  2. Live Session 22-03-2024 Video
  3. Live Session 12-04-2024 Video
  4. Live Session 07-03-2025 Video
  5. Live Session 27-03-2025 Video

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IIT Bombay
Department
Mathematics
Teacher
Prof. G. K Srinivasan
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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