I always thought that Fourier was the first to come up with the idea of expressing general functions as infinite sums of sines and cosines. Apparently this isn’t true.

The idea that various functions can be described in terms of Fourier series … was for the first time proposed by Daniel Bernoulli (1700–1782) to solve the one-dimensional wave equation (the equation of motion of a string) about 50 years before Fourier. … However, no one contemporaneous to D. Bernoulli accepted the idea as a general method, and soon the study was forgotten.

Source: The Nonlinear World

Perhaps Fourier’s name stuck to the idea because he developed it further than Bernoulli did.

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That is interesting.

Interesting that you mention earlier versions of Fourier series on the day after Nicolaus Copernicus birthday. I remember hearing years ago that one point of view on the earlier Ptolemaic system of representing orbits with cycles and epicycles (and epicycles on epicycles, …) was that it was effectively doing a kind of Fourier series expansion to describe the motion without *really* knowing what they were doing.

My limited understanding of the history here is that many people (e.g. Euler) knew about the method of Fourier analysis, but did not believe that Fourier series would be generally applicable except to specialized functions.

It’s interesting that the theory of Fourier series is so different from the theory of Taylor series. At first glance they don’t seem that different, just a different choice of basis. But they also have different notions of convergence, and that leads to different theories. Taylor series converge uniformly in a disk, but Fourier series converge naturally in L

^{2}norm.Things get difficult when you start asking about how Fourier series behave with respect to other modes of convergence. For example, it wasn’t until 1966 that Carleson proved that Fourier series converge pointwise almost everywhere.

Indeed is an amazing history. I recommend the short essay “Controversy in science: on the Ideas of Daniel Bernoulli and René Thom” by E.C. Zeeman. It describes the whole controversy about the Fourier Series discovery, you can find it on line.

Should this be added to the examples of Stigler’s Law?

http://en.wikipedia.org/wiki/List_of_examples_of_Stigler%27s_law

You can find an article by Giovanni Gallavotti describing, among other things, the connection between epicycles and Fourier series at

http://www.lincei.it/pubblicazioni/rendicontiFMN/rol/pdf/M2001-02-06.pdf

As Federico already mentioned, perhaps the earliest account for the ideas come from the astronomers of the ancient Greek, as they tried to model the planetary motion as a succession of epicycles. That is, they were determining the terms of the Fourier series of the path they were observing on the sky. More at http://math.stackexchange.com/questions/1002/fourier-transform-for-dummies and at http://en.wikipedia.org/wiki/Deferent_and_epicycle .