Episode Details

This post records what I've learned while studying a bit of Fourier analysis. I used this PDF, which is the lecture notes for this Stanford course. The only thing in here that is really changed from there is the derivation of the Fourier transform, where I tried to explain the way I made sense of it. (That explanation may or may not make sense.)

Fourier Series

Fourier analysis starts with the study of periodic functions. The fundamental periodic function is the complex exponential , which comes up as the solution of the harmonic oscillator equation and many other places. In the complex plane, this function starts on the horizontal axis at , then spins counterclockwise through the unit circle at an angular velocity of . (Or clockwise at that speed if is negative.)

Of course, most of our functions are real-valued, not complex. Fourier himself developed his theory of Fourier series and transforms using sines and cosines. But we have since found that the equations come out much neater if you write them in terms of complex exponentials. So we will write sines and cosines in terms of the complex exponential:

And we'll take Fourier series and Fourier [...]

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Outline:

(00:30) Fourier Series

(06:03) The Fourier Transform

(11:49) Using the Fourier Transform

(15:25) A More General Definition

(20:50) Ш and the Sampling Theorem

(27:03) The Discrete Fourier Transform

The original text contained 9 footnotes which were omitted from this narration.

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First published:
May 26th, 2026

Source:
https://www.lesswrong.com/posts/qZCGcmqM69x9r7D85/notes-on-fourier-analysis

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Narrated by TYPE III AUDIO.

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Images from the article:

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