Calculus
Mathematical Analysis
Numerical Series Expansion

# What is the history of fourier series?

###### Wiki User

It is quite complicated, and starts before Fourier. Trigonometric series arose in problems connected with Astronomy in the 1750s, and were tackled by Euler and others. In a different context, they arose in connection with a vibrating string (e.g. a violin string) and solutions of the wave equation.

Still in the 1750s, a controversy broke out as to what curves could be represented by trigonometric series and whether every solution to the wave equation could be represented as the sum of a trigonometric series; Daniel Bernoulli claimed that every solution could be so represented and Euler claimed that arbitrary curves could not necessarily be represented. The argument rumbled on for 20 years and dragged in other people, including Laplace. At that time the concepts were not available to settle the problem.

Fourier worked on the heat equation (controlling the diffusion of heat in solid bodies, for example the Earth) in the early part of the 19th century, including a major paper in 1811 and a book in 1822. Fourier had a broader notion of function than the 18th-century people, and also had more convincing examples.

Fourier's work was criticised at the time, and his insistence that discontinuous functions could be represented by trigonometric series contradicted a theorem in a textbook by the leading mathematician of the time, Cauchy.

Nonetheless Fourier was right; Cauchy (and Fourier, and everyone else at that time) was missing the idea of uniform convergence of a series of functions. Fourier's work was widely taken up, and also the outstanding problems (just which functions can be represented by Fourier series?; how different can two functions be if they have the same Fourier series?) were slowly solved.

Source: Morris Kline, Mathematical Thought from Ancient to Modern Times, Oxford University Press, 1972, pages 478-481, 502-514, 671-678,and 964.

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## Related Questions

what are the limitations of forier series over fourier transform

yes a discontinuous function can be developed in a fourier series

Fourier series and the Fourier transform

Fourier series is series which help us to solve certain physical equations effectively

Fourier series is the sum of sinusoids representing the given function which has to be analysed whereas discrete fourier transform is a function which we get when summation is done.

The fourier series of a sine wave is 100% fundamental, 0% any harmonics.

no every function cannot be expressed in fourier series... fourier series can b usd only for periodic functions.

Yes, a Fourier series can be used to approximate a function with some discontinuities. This can be proved easily.

James Ward Brown has written: 'Fourier series and boundary value problems' -- subject(s): Boundary value problems, Fourier series, Functions, Orthogonal, Orthogonal Functions 'Fourier series and boundary value problems' -- subject(s): Boundary value problems, Fourier series, Orthogonal Functions

Joseph Fourier was the French mathematician and physicist after whom Fourier Series, Fourier's Law, and the Fourier Transform were named. He is commonly credited with discovering the greenhouse effect.

A Fourier series is a set of harmonics at frequencies f, 2f, 3f etc. that represents a repetitive function of time that has a period of 1/f. A Fourier transform is a continuous linear function. The spectrum of a signal is the Fourier transform of its waveform. The waveform and spectrum are a Fourier transform pair.

The Fourier series can be used to represent any periodic signal using a summation of sines and cosines of different frequencies and amplitudes. Since sines and cosines are periodic, they must form another periodic signal. Thus, the Fourier series is period in nature. The Fourier series is expanded then, to the complex plane, and can be applied to non-periodic signals. This gave rise to the Fourier transform, which represents a signal in the frequency-domain. See links.

S. K. Suslov has written: 'An introduction to basic Fourier series' -- subject(s): Fourier series

Tatsuo Kawata has written: 'Fourier analysis in probability theory' -- subject(s): Fourier series, Fourier transformations, Probabilities

when we have need to know the temperature in a bar about any distance we can use fourier series to know that and then we can apply sufficient temperature.

An aperiodic signal cannot be represented using fourier series because the definition of fourier series is the summation of one or more (possibly infinite) sine wave to represent a periodicsignal. Since an aperiodic signal is not periodic, the fourier series does not apply to it. You can come close, and you can even make the summation mostly indistinguishable from the aperiodic signal, but the math does not work.

The Fourier series is a specific type of infinite mathematical series involving trigonometric functions that are used in applied mathematics. It makes use of the relationships of the sine and cosine functions.

A Fourier series is a series of sine and cosine harmonics of a particular frequency. For example sinf+icosf + 3 sin2f+ 5icos2f... where the successive terms are multiples of the fundamental frequency f. It is typical ( but as far as I know not required) that complex numbers are used. A Fourier transform converts a time domain wave form (like a sound wave) into the coefficients of the corresponding Fourier series. A DFT is a digital approximation to a Fourier transform, usually using something like the Cooley-Tuckey Fast Fourier Transform (FFT) for efficiency. The underlying Fourier theorem is something like: Every bounded periodic continuous (needed to avoid Gibbs) function , or wave form, can be written as the sum of its Fourier series. i.e. It is a sum of sines and cosines In otherwords, you take a wave form in the time domain like a sound wave and break it into its components (various frequencies) by the Fourier Transform. The results of the Transform are the coefficients of the Fourier series. The wave form of a voice converted to components (and perhaps a little more) is a voiceprint.

You can graph both with Energy on the y-axis and frequency on the x. Such a frequency domain graph of a fourier series will be discrete with a finite number of values corresponding to the coefficients a0, a1, a2, ...., b1, b2,... Also, the fourier series will have a limited domain corresponding to the longest period of your original function. A fourier transforms turns a sum into an integral and as such is a continuous function (with uncountably many values) over the entire domain (-inf,inf). Because the frequency domain is unrestricted, fourier transforms can be used to model nonperiodic functions as well while fourier series only work on periodic ones. Series: discrete, limited domain Transform: continuous, infinite domain.

J. Ray Hanna has written: 'Fourier series and integrals of boundary value problems' -- subject(s): Boundary value problems, Fourier series

Jean-Pierre Kahane has written: 'Fourier series and wavelets' 'Series de Fourier absolument convergentes' 'Some random series of functions' -- subject(s): Stochastic processes, Series, Functions, Random variables

I think the following Wikipedia link on Fourier Series (see related links below), has the information that you're looking for.

Consider a periodic function, generally defined by f(x+t) = f(x) for some t. Any periodic function can be written as an infinite sum of sines and cosines. This is called a Fourier series.

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