Questions tagged [harmonic-analysis]

Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms.

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What did Fourier mean by stating that every function can be decomposed into sine and cosine functions?

Fourier stated that every function can be decomposed into sine and cosine functions. Was he referring to periodic functions only? To a certain class only? I ask, because it seems clear (at least to me)...
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Was Fourier inspired by Ptolemy?

Ptolemy invented a system to describe the periodic motion of the planets by epicycles. Fourier did something similar for periodic motion in mechanics. Every such motion can be thought of as ...
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What does the Fourier transform have to do with heat?

For example the current version of the Fourier analysis article on Wikipedia says the study is: […] named after Joseph Fourier, who showed that representing a function as a sum of trigonometric ...
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How did Fourier determine the coefficients of Fourier series?

I was reading a chapter of Fourier's seminal work "Analytic Theory of Heat". The third chapter of this book was translated by the famous Stephen Hawking in his book "God created the ...
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Has anyone explored Ptolemy's epicycles as an early form of Fourier analysis?

Whilst researching science in the ancient world, I came across an observation, which unfortunately I did not make a note of, and so cannot credit, that Ptolemy's epicycles were an early form of ...
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How was Fourier analysis important to the development of set theory?

I recently read the following quote (unfortunately, I copied it down without attribution): You may be surprised to know that Fourier analysis played a role in the early development of set theory. In ...
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Origin of the Fourier transform (1878)

I located Joseph Fourier's book, Analytical Theory of Heat (1878), but at first glance it looks like it is all about heat. What did Fourier call the Fourier transform? When did he first use it?
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Substantiating claimed Fourier quote about “an arbitrarily capricious graph”

The following quote (in English) is fairly widely attributed to Fourier, but I can't substantiate it. An arbitrary function, continuous or with discontinuities, defined in a finite interval by an ...
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On the origin of the concept of aliasing & the Discrete Fourier Transform frequency axis

The development of the fast Fourier transform (FFT) is attributed to Cooley & Tukey, both of whom have written a lot about its historical development. However, I am searching for early ...
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Who came up with the convolution theorem?

I am looking for the earliest reference which proposed the convolution theorem which is often utilized in signal processing (i.e., convolution becomes multiplication in the Fourier domain). The ...
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550 views

What are some good references elucidating the discovery/creation of Fourier Series?

I've always grappled with anything related to Fourier since my undergrad days. Recently, when revisiting why I learned what I did, I discovered how Fourier's desire to understand the flow of heat ...
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Dirichlet's Proof of the Convergence of Fourier Series

Where can I find Dirichlet's proof of the convergence of Fourier series?
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First evaluation of $\sum_{n \geq 1} 1/n^2$ by Fourier series

There are many ways to evaluate $\sum_{n \geq 1} 1/n^2$ as $\pi^2/6$, including multiple solutions using Fourier series. A colleague asked me who was the first person to use Fourier series (or Fourier ...