Fourier transforms 

Continuous Fourier transform 
Fourier series 
Discretetime Fourier transform 
Discrete Fourier transform 
Discrete Fourier transform over a ring 
Fourier analysis 
Related transforms 
In mathematics, a Fourier series ^{[1]} is a way to represent a function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The discretetime Fourier transform is a periodic function, often defined in terms of a Fourier series. The Ztransform, another example of application, reduces to a Fourier series for the important case z=1. Fourier series are also central to the original proof of the NyquistShannon sampling theorem. The study of Fourier series is a branch of Fourier analysis.
The Fourier series is named in honour of JeanBaptiste Joseph Fourier (17681830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli.^{[nb 1]} Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides (Treatise on the propagation of heat in solid bodies), and publishing his Théorie analytique de la chaleur (Analytical theory of heat) in 1822. The Mémoire introduced Fourier analysis, specifically Fourier series. Through Fourier's research the fact was established that an arbitrary (continuous)^{[2]} function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before the French Academy.^{[3]} Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on deferents and epicycles.
The heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series.
From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. Later, Peter Gustav Lejeune Dirichlet^{[4]} and Bernhard Riemann^{[5]}^{[6]}^{[7]} expressed Fourier's results with greater precision and formality.
Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids. The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics,^{[8]}thinwalled shell theory,^{[9]} etc.
In this section, s(x) denotes a function of the real variable x, and s is integrable on an interval [x_{0}, x_{0} + P], for real numbers x_{0} and P. We will attempt to represent s in that interval as an infinite sum, or series, of harmonically related sinusoidal functions. Outside the interval, the series is periodic with period P (frequency 1/P). It follows that if s also has that property, the approximation is valid on the entire real line. We can begin with a finite summation (or partial sum):
is a periodic function with period P. Using the identities:
we can also write the function in these equivalent forms:
where:
The inverse relationships between the coefficients are:
When the coefficients (known as Fourier coefficients) are computed as follows:^{[10]}

approximates on and the approximation improves as N > ?. The infinite sum, is called the Fourier series representation of
Both components of a complexvalued function are realvalued functions that can be represented by a Fourier series. The two sets of coefficients and the partial sum are given by:
This is the same formula as before except c_{n} and c_{n} are no longer complex conjugates. The formula for c_{n} is also unchanged:
In engineering applications, the Fourier series is generally presumed to converge everywhere except at discontinuities, since the functions encountered in engineering are more well behaved than the ones that mathematicians can provide as counterexamples to this presumption. In particular, if s is continuous and the derivative of s(x) (which may not exist everywhere) is square integrable, then the Fourier series of s converges absolutely and uniformly to s(x).^{[11]} If a function is squareintegrable on the interval [x_{0}, x_{0}+P], then the Fourier series converges to the function at almost every point. Convergence of Fourier series also depends on the finite number of maxima and minima in a function which is popularly known as one of the Dirichlet's condition for Fourier series. See Convergence of Fourier series. It is possible to define Fourier coefficients for more general functions or distributions, in such cases convergence in norm or weak convergence is usually of interest.
Another visualisation of an approximation of a square wave by taking the first 1, 2, 3 and 4 terms of its Fourier series. (An interactive animation can be seen here)
We now use the formula above to give a Fourier series expansion of a very simple function. Consider a sawtooth wave
In this case, the Fourier coefficients are given by
It can be proven that Fourier series converges to s(x) at every point x where s is differentiable, and therefore:

When x = ?, the Fourier series converges to 0, which is the halfsum of the left and rightlimit of s at x = ?. This is a particular instance of the Dirichlet theorem for Fourier series.
This example leads us to a solution to the Basel problem.
The Fourier series expansion of our function in Example 1 looks more complicated than the simple formula s(x) = x/?, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the heat equation. For example, consider a metal plate in the shape of a square whose side measures ? meters, with coordinates (x, y) ? [0, ?] × [0, ?]. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by y = ?, is maintained at the temperature gradient T(x, ?) = x degrees Celsius, for x in (0, ?), then one can show that the stationary heat distribution (or the heat distribution after a long period of time has elapsed) is given by
Here, sinh is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of Eq.1 by sinh(ny)/sinh(n?). While our example function s(x) seems to have a needlessly complicated Fourier series, the heat distribution T(x, y) is nontrivial. The function T cannot be written as a closedform expression. This method of solving the heat problem was made possible by Fourier's work.
Another application of this Fourier series is to solve the Basel problem by using Parseval's theorem. The example generalizes and one may compute ?(2n), for any positive integer n.
The notation c_{n} is inadequate for discussing the Fourier coefficients of several different functions. Therefore, it is customarily replaced by a modified form of the function (s, in this case), such as or S, and functional notation often replaces subscripting:
In engineering, particularly when the variable x represents time, the coefficient sequence is called a frequency domain representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.
Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:
where f represents a continuous frequency domain. When variable x has units of seconds, f has units of hertz. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of 1/P, which is called the fundamental frequency. can be recovered from this representation by an inverse Fourier transform:
The constructed function S(f) is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.^{[nb 2]}
" 
Multiplying both sides by , and then integrating from to yields: 
" 
 Joseph Fourier, Mémoire sur la propagation de la chaleur dans les corps solides. (1807)^{[12]}^{[nb 3]} 
This immediately gives any coefficient a_{k} of the trigonometrical series for ?(y) for any function which has such an expansion. It works because if ? has such an expansion, then (under suitable convergence assumptions) the integral
can be carried out termbyterm. But all terms involving for vanish when integrated from 1 to 1, leaving only the kth term.
In these few lines, which are close to the modern formalism used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli and Gauss, Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis.
When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.^{[]}
Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available at the time Fourier completed his original work. Fourier originally defined the Fourier series for realvalued functions of real arguments, and using the sine and cosine functions as the basis set for the decomposition.
Many other Fourierrelated transforms have since been defined, extending the initial idea to other applications. This general area of inquiry is now sometimes called harmonic analysis. A Fourier series, however, can be used only for periodic functions, or for functions on a bounded (compact) interval.
We can also define the Fourier series for functions of two variables x and y in the square [?, ?] × [?, ?]:
Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in image compression. In particular, the jpeg image compression standard uses the twodimensional discrete cosine transform, which is a Fourier transform using the cosine basis functions.
The threedimensional Bravais lattice is defined as the set of vectors of the form:
where n_{i} are integers and a_{i} are three linearly independent vectors. Assuming we have some function, f(r), such that it obeys the following condition for any Bravais lattice vector R: f(r) = f(r + R), we could make a Fourier series of it. This kind of function can be, for example, the effective potential that one electron "feels" inside a periodic crystal. It is useful to make a Fourier series of the potential then when applying Bloch's theorem. First, we may write any arbitrary vector r in the coordinatesystem of the lattice:
where a_{i} = a_{i}.
Thus we can define a new function,
This new function, , is now a function of threevariables, each of which has periodicity a_{1}, a_{2}, a_{3} respectively:
If we write a series for g on the interval [0, a_{1}] for x_{1}, we can define the following:
And then we can write:
Further defining:
We can write g once again as:
Finally applying the same for the third coordinate, we define:
We write g as:
Rearranging:
Now, every reciprocal lattice vector can be written as , where l_{i} are integers and g_{i} are the reciprocal lattice vectors, we can use the fact that to calculate that for any arbitrary reciprocal lattice vector K and arbitrary vector in space r, their scalar product is:
And so it is clear that in our expansion, the sum is actually over reciprocal lattice vectors:
where
Assuming
we can solve this system of three linear equations for x, y, and z in terms of x_{1}, x_{2} and x_{3} in order to calculate the volume element in the original cartesian coordinate system. Once we have x, y, and z in terms of x_{1}, x_{2} and x_{3}, we can calculate the Jacobian determinant:
which after some calculation and applying some nontrivial crossproduct identities can be shown to be equal to:
(it may be advantageous for the sake of simplifying calculations, to work in such a cartesian coordinate system, in which it just so happens that a_{1} is parallel to the x axis, a_{2} lies in the xy plane, and a_{3} has components of all three axes). The denominator is exactly the volume of the primitive unit cell which is enclosed by the three primitivevectors a_{1}, a_{2} and a_{3}. In particular, we now know that
We can write now h(K) as an integral with the traditional coordinate system over the volume of the primitive cell, instead of with the x_{1}, x_{2} and x_{3} variables:
And C is the primitive unit cell, thus, is the volume of the primitive unit cell.
In the language of Hilbert spaces, the set of functions is an orthonormal basis for the space L^{2}([?, ?]) of squareintegrable functions on [?, ?]. This space is actually a Hilbert space with an inner product given for any two elements f and g by
The basic Fourier series result for Hilbert spaces can be written as
This corresponds exactly to the complex exponential formulation given above. The version with sines and cosines is also justified with the Hilbert space interpretation. Indeed, the sines and cosines form an orthogonal set:
(where ?_{mn} is the Kronecker delta), and
furthermore, the sines and cosines are orthogonal to the constant function 1. An orthonormal basis for L^{2}([?,?]) consisting of real functions is formed by the functions 1 and cos(nx), sin(nx) with n = 1, 2,... The density of their span is a consequence of the StoneWeierstrass theorem, but follows also from the properties of classical kernels like the Fejér kernel.
We say that f belongs to if f is a 2?periodic function on R which is k times differentiable, and its kth derivative is continuous.
One of the interesting properties of the Fourier transform which we have mentioned, is that it carries convolutions to pointwise products. If that is the property which we seek to preserve, one can produce Fourier series on any compact group. Typical examples include those classical groups that are compact. This generalizes the Fourier transform to all spaces of the form L^{2}(G), where G is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products. The Fourier series exists and converges in similar ways to the [?,?] case.
An alternative extension to compact groups is the PeterWeyl theorem, which proves results about representations of compact groups analogous to those about finite groups.
If the domain is not a group, then there is no intrinsically defined convolution. However, if X is a compact Riemannian manifold, it has a LaplaceBeltrami operator. The LaplaceBeltrami operator is the differential operator that corresponds to Laplace operator for the Riemannian manifold X. Then, by analogy, one can consider heat equations on X. Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the LaplaceBeltrami operator as a basis. This generalizes Fourier series to spaces of the type L^{2}(X), where X is a Riemannian manifold. The Fourier series converges in ways similar to the [?, ?] case. A typical example is to take X to be the sphere with the usual metric, in which case the Fourier basis consists of spherical harmonics.
The generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups. However, there is a straightforward generalization to Locally Compact Abelian (LCA) groups.
This generalizes the Fourier transform to L^{1}(G) or L^{2}(G), where G is an LCA group. If G is compact, one also obtains a Fourier series, which converges similarly to the [?, ?] case, but if G is noncompact, one obtains instead a Fourier integral. This generalization yields the usual Fourier transform when the underlying locally compact Abelian group is R.
An important question for the theory as well as applications is that of convergence. In particular, it is often necessary in applications to replace the infinite series by a finite one,
This is called a partial sum. We would like to know, in which sense does f_{N}(x) converge to f(x) as N > ?.
We say that p is a trigonometric polynomial of degree N when it is of the form
Note that f_{N} is a trigonometric polynomial of degree N. Parseval's theorem implies that
Theorem. The trigonometric polynomial f_{N} is the unique best trigonometric polynomial of degree N approximating f(x), in the sense that, for any trigonometric polynomial p ? f_{N} of degree N, we have
where the Hilbert space norm is defined as:
Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.
Theorem. If f belongs to L^{2}([?, ?]), then f_{?} converges to f in L^{2}([?, ?]), that is, converges to 0 as N > ?.
We have already mentioned that if f is continuously differentiable, then is the nth Fourier coefficient of the derivative f?. It follows, essentially from the CauchySchwarz inequality, that f_{?} is absolutely summable. The sum of this series is a continuous function, equal to f, since the Fourier series converges in the mean to f:
Theorem. If , then f_{?} converges to f uniformly (and hence also pointwise.)
This result can be proven easily if f is further assumed to be C^{2}, since in that case tends to zero as n > ?. More generally, the Fourier series is absolutely summable, thus converges uniformly to f, provided that f satisfies a Hölder condition of order ? > ½. In the absolutely summable case, the inequality proves uniform convergence.
Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at x if f is differentiable at x, to Lennart Carleson's much more sophisticated result that the Fourier series of an L^{2} function actually converges almost everywhere.
These theorems, and informal variations of them that don't specify the convergence conditions, are sometimes referred to generically as "Fourier's theorem" or "the Fourier theorem".^{[14]}^{[15]}^{[16]}^{[17]}
Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. For example, the Fourier series of a continuous Tperiodic function need not converge pointwise. The uniform boundedness principle yields a simple nonconstructive proof of this fact.
In 1922, Andrey Kolmogorov published an article titled "Une série de FourierLebesgue divergente presque partout" in which he gave an example of a Lebesgueintegrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere (Katznelson 1976).
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(help) 2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's work Théorie Analytique de la Chaleur, originally published in 1822.This article incorporates material from example of Fourier series on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.