Joshua Arnold

Fourier Analysis - Spring 2024 DRP

Materials

Fourier analysis: An introduction by Elias M. Stein & Rami Shakarchi

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Fourier Series (Fourier Analysis on a Circle)

If f is an integrable periodic function given on an interval [a,b] of some length L, then the nth Fourier coefficient of f is defined as :

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Similarly if given the Fourier coefficients we can reproduce f as :

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Fourier Transform (Analysis for Functions on R)

Extending the Fourier Series to all of R and for non-periodic functions such that f ∈ S(R), we define the Fourier transform for ξ ∈ R by

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The Fourier Inversion formula allows us to recover f(x) by

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Fourier Transform Application to Heat Equation

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Convolutions

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*The main idea is that Fourier Transforms turn convolutions into multiplication

Fourier Transform Application to Heat Equation

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Fourier Transform (Analysis for Functions R^d)

The Fourier transform of a Schwartz function f is defined by:

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Similarly if f ∈ S(R^d), then the inverse transform is provided by

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Discrete Fourier Transform

Letting a continuous function f(x) be the source of the data and N be the number of samples denoted by f[0], f[1], f[2], …, f[k], f[N-1], the once continuous integrand of the F.T now exists solely at these sample points. So our DFT is defined as:

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The Inverse DFT is then defined as:

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*Treats data as if periodic ( f(N) to f(2N-1) same as f(0) to f(N-1) )

Fast Fourier Transform

Highly efficient modification of the DFT developed in the mid-60’s. This gets rid of redundant calculations.

�Rewrite:

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Ex). N=8

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Demonstrations!

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