Fourier Transforms

Fourier transforms are extremely useful in Signal processing and for solving partial differential equations. Thus it forms an important component in any engineering education. We shall introduce the basic space on which it is defined and examine its basic properties. We shall derive the Fourier transform of the Gaussian and look at some applications.

Relevant Course : Partial differential equations and electrical engineering.

Relevant Department : Mathematics

Relevant Semester : 3

Pre- requisite : ODEs and basic one variable calculus.

Schedule for Lecture Delivery

Session 1 :

Session 2 : 

Session 3 : 

Fourier Transforms and Partial Differential Equations

Fourier transforms and partial differential equations

Recall that in basic ODE  theory where one studies equations with constant coefficients, special solutions were sought in the form

is called the Fourier transform of f(t). There are several conventions and we follow the one that is common in PDEs for example seep. 213 of G. B. Folland, Fourier analysis and its applications.

Exercises:

Exercises:

Section

Section 1

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Ref video on Fourier Transforms and partial differential equations

Section 2

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Ref video on Definition of Fourier Transforms

Section 3

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Ref video on Definition of Fourier Transforms Contd..

Section 4

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Ref video on Definition of Fourier Transforms contd

Section 5

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Ref video on Fourier Transform of the Gaussian

Section 6

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The Schwartz space S of rapidly decreasing functions

Section 7

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Ref video on Properties of S

Section 8

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Ref video on Hermite's ODE and Hermite polynomials

Hermites ODE and Hermite

Lecture II : Hermite's ODE and Hermite polynomials

Slides for Lecture 2

Slides for Lecture 2

Fourier Transforms of the Gaussian ( Part I)

Fourier Transforms of the Gaussian ( Part II)

In Class Questions & Hermite's ODE & Hermite's Polynomial again

Fourier Inversion Theorem (Part I)

Fourier Inversion Theorem (Part II )

Ramanujan's Formula (Part I)

Ramanujan's Formula (Part II)

Radial Functions - The Bessel Transform

Lecture - III: Radial functions - the Bessel transform

More generally in even space dimensions it reduces to a Bessel transform and in odd dimension a "sine transform".

38. Determine the radius of convergence of the power series solution of Airy's equation. We have now both the series solution as well as the integral representation. Which one would you think is more useful?

So the situation has decidedly improved! One more integration by parts would now prove the assertion.

Slides for Lecture 3

Slides for Lecture 3

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The Parseval Formula

The Application of Heat Equation

D'Alemberts Solution of the Wave Equation

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Radial Functions-The Bessel Transform


Principle of Equipartioning of Energy for Wave Equation

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Airys Function

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