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Fourier transforms 

Continuous Fourier transform 
Fourier series 
Discrete Fourier transform 
Discretetime Fourier transform 

In mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate.
The heat equation is a partial differential equation. Prior to Fourier's work, there was no known solution to the heat equation in a general situation, 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.
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. The basic results are very easy to understand using the modern theory.
The Fourier series has many applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics^{[1]}, etc.
Fourier series is named in honour of 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. He applied this technique to find the solution of the heat equation, publishing his initial results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides and 1811, and publishing his Théorie analytique de la chaleur in 1822.
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, Dirichlet and Riemann expressed Fourier's results with greater precision and formality.
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“ 
Multiplying both sides by , and then integrating from y = − 1 to y = + 1 yields: 
” 
—Joseph Fourier, Mémoire sur la propagation de la chaleur dans les corps solides, pp. 218–219.^{[2]} 
In these few lines, which are surprisingly close to the modern formalism used in Fourier series, Fourier unwittingly 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 arbitrary functions. 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 his paper in 1807, 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.
In this section, ƒ(x) denotes a function of the real variable x. This function is usually taken to be periodic, of period 2π, which is to say that ƒ(x + 2π) = ƒ(x), for all real numbers x. We will attempt to write such a function as an infinite sum, or series of simpler 2π–periodic functions. We will start by using an infinite sum of sine and cosine functions on the interval [−π, π], as Fourier did (see the quote above), and we will then discuss different formulations and generalizations.
For a periodic function ƒ(x) that is integrable on [−p, p], the numbers
and
are called the Fourier coefficients of ƒ. One introduces the partial sums of the Fourier series for ƒ, often denoted by
The partial sums for ƒ are trigonometric polynomials. One expects that the functions S_{N} ƒ approximate the function ƒ, and that the approximation improves as N tends to infinity. The infinite sum
is called the Fourier series of ƒ.
The Fourier series does not always converge, and even when it does converge for a specific value x_{0} of x, the sum of the series at x_{0} may differ from the value ƒ(x_{0}) of the function. It is one of the main questions in harmonic analysis to decide when Fourier series converge, and when the sum is equal to the original function. If a function is squareintegrable on the interval [−π, π], then the Fourier series converges to the function at almost every point. 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, the Fourier series converges absolutely and uniformly to ƒ(x) whenever the derivative of ƒ(x) (which may not exist everywhere) is square integrable.^{[3]} 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.
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 proved that the Fourier series converges to ƒ(x) at every point x where ƒ is differentiable, and therefore:

( 
When x = π, the Fourier series converges to 0, which is the halfsum of the left and rightlimit of ƒ at x = π. This is a particular instance of the Dirichlet theorem for Fourier series.
One notices that the Fourier series expansion of our function looks much less simple than the formula ƒ(x) = x, and so it is not immediately apparent why one would need this Fourier series. While there are many applications, we cite Fourier's motivation of solving the heat equation. For example, consider a metal plate in the shape of a square whose side measures π meters, with coordinates (x, y) ∈ [0, π] × [0, π]. 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 = π, is maintained at the temperature gradient T(x, π) = x degrees Celsius, for x in (0, π), 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(nπ). While our example function f(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 only 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 ζ(2n), for any positive integer n.
We can use Euler's formula,
where i is the imaginary unit, to give a more concise formula:
The Fourier coefficients are then given by:
The Fourier coefficients a_{n}, b_{n}, c_{n} are related via
and
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 ƒ (in this case), such as F or and functional notation often replaces subscripting. Thus:
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.
The following formula, with appropriate complexvalued coefficients G[n], is a periodic function with period τ on all of R:
If a function is squareintegrable in the interval [a, a + τ], it can be represented in that interval by the formula above. If g(x) is integrable, then the Fourier coefficients are given by:
Note that if the function to be represented is also τperiodic, then a is an arbitrary choice. Two popular choices are a = 0, and a = −τ/2.
Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:
where variable ƒ represents a continuous frequency domain. When variable x has units of seconds, ƒ has units of hertz. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of 1/τ, which is called the fundamental frequency. The original g(x) can be recovered from this representation by an inverse Fourier transform:
The function G(ƒ) is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent.^{[4]}
We can also define the Fourier series for functions of two variables x and y in the square [−π, π]×[−π, π]:
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.
In the language of Hilbert spaces, the set of functions is an orthonormal basis for the space L^{2}([ − π,π]) of squareintegrable functions of [ − π,π]. This space is actually a Hilbert space with an inner product given 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 δ_{mn} is the Kronecker delta), and
furthermore, the sines and cosines are orthogonal to the constant function 1. An orthonormal basis for L^{2}([−π, π]) consisting of real functions is formed by the functions 1, and √2 cos(n x), √2 sin(n x) for n = 1, 2,... The density of their span is a consequence of the Stone–Weierstrass theorem, but follows also from the properties of classical kernels like the Fejér kernel.
We say that ƒ belongs to if ƒ is a 2πperiodic function on R which is k times differentiable, and its kth derivative is continuous.
There are many possible avenues for generalizing Fourier series. The study of Fourier series and its generalizations is called harmonic analysis.
One can extend the notion of Fourier coefficients to functions which are not squareintegrable, and even to objects which are not functions. This is very useful in engineering and applications because we often need to take the Fourier series of a periodic repetition of a Dirac delta function. The Dirac delta δ is not actually a function; still, it has a Fourier transform and its periodic repetition has a Fourier series:
This generalization to distributions enlarges the domain of definition of the Fourier transform from L^{2}([−π, π]) to a superset of L^{2}. The Fourier series converges weakly.
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 [−π, π] case.
An alternative extension to compact groups is the Peter–Weyl 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 [−π, π] 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 straightfoward 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 [−π, π] 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 .
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 (S_{N} ƒ)(x) converge to ƒ(x) as N tends to infinity.
We say that p is a trigonometric polynomial of degree N when it is of the form
Note that S_{N} ƒ is a trigonometric polynomial of degree N. Parseval's theorem implies that
Theorem. The trigonometric polynomial S_{N} ƒ is the unique best trigonometric polynomial of degree N approximating ƒ(x), in the sense that, for any trigonometric polynomial of degree N, we have
Here, the Hilbert space norm is
Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.
Theorem. If ƒ belongs to L^{2}([−π, π]), then the Fourier series converges to ƒ in L^{2}([−π, π]), that is, converges to 0 as N goes to infinity.
We have already mentioned that if ƒ is continuously differentiable, then is the nth Fourier coefficient of the derivative ƒ′. It follows, essentially from the CauchySchwarz inequality, that the Fourier series of ƒ is absolutely summable. The sum of this series is a continuous function, equal to ƒ, since the Fourier series converges in the mean to ƒ:
Theorem. If , then the Fourier series converges to ƒ uniformly (and hence also pointwise.)
This result can be proven easily if ƒ is further assumed to be C^{2}, since in that case tends to zero as . More generally, the Fourier series is absolutely summable, thus converges uniformly to ƒ, provided that ƒ satisfies a Hölder condition of order α > 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 ƒ 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 specifty the convergence conditions, are sometimes referred to generically as "Fourier's theorem" or "the Fourier theorem".^{[5]}^{[6]}^{[7]}^{[8]}
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.
In 1922, Andrey Kolmogorov published an article entitled "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).
This article incorporates material from example of Fourier series on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. The weights, or coefficients, of the modes, are a onetoone mapping of the original function. Generalizations include generalized Fourier series and other expansions over orthonormal bases.
Fourier series serve many useful purposes, as manipulation and conceptualization of the modal coefficients are often easier than with the original function. Areas of application include electrical engineering, vibration analysis, acoustics, optics, signal and image processing, and data compression. Using the tools and techniques of spectroscopy, for example, astronomers can deduce the chemical composition of a star by analyzing the frequency components, or spectrum, of the star's emitted light. Similarly, engineers can optimize the design of a telecommunications system using information about the spectral components of the data signal that the system will carry. See also spectrum analyzer.
The Fourier series is named after the French scientist and mathematician Joseph Fourier, who used them in his influential work on heat conduction, Théorie Analytique de la Chaleur (The Analytical Theory of Heat), published in 1822.
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Given a complexvalued function f of real argument t, f: R → C, where f(t) is piecewise smooth and continuous, periodic with period T, and squareintegrable over the interval from t_{1} to t_{2} of length T, that is,
where
The Fourier series expansion of f is
where, for any nonnegative integer n,
Equivalently, in complex exponential form,
where:
For a formal justification, see Modern derivation of the Fourier coefficients below.
In the special case where the period T = 2π, we have
In this case, the Fourier series expansion reduces to a particularly simple form:
where
for any nonnegative integer n.
or, equivalently:
where
The form for period T can be easily derived from the canonical one with the change of variable defined by . Therefore, both formulations are equivalent. However, the form for period T is used in most practical cases because it is directly applicable. For the theory, the canonical form is preferred because it is more elegant and easier to interpret mathematically, as will later be seen.
Let f be periodic of period 2π, with f(x) = x for x from −π to π. Note that this function is a periodic version of the identity function.
We will compute the Fourier coefficients for this function.
Notice that a_{n} are 0 because the are odd functions. Hence the Fourier series for this function is:
One application of this Fourier series is to compute the value of the Riemann zeta function at s = 2; by Parseval's theorem, we have:
which yields: .
The wave equation governs the motion of a vibrating string, which may be fastened down at its endpoints. The solution of this problem requires the trigonometric expansion of a general function f that vanishes at the endpoints of an interval x=0 and x=L. The Fourier series for such a function takes the form
where
Vibrations of air in a pipe that is open at one end and closed at the other are also described by the wave equation. Its solution requires expansion of a function that vanishes at x = 0 and whose derivative vanishes at x=L. The Fourier series for such a function takes the form
where
Fourier series have a kinematic interpretation. Indeed, the function can be seen as the movement of an object on a plane (t would then represent time). Since f is complexvalued, we can write
for realvalued functions u and v. In this form, we can interpret f as a sum of horizontal and vertical translations.
From time t to time t + dt, where dt is a very small incremental period, the object moves from the point to the point , which corresponds to an infinitesimal translation in space by the vector . As a result, we can write f as:
Now instead of seeing f as a sum of infinitesimal translations, we can see it as an infinite sum of rotations of different radii. This interpretation is convenient, in particular when the movement is periodic.
Let χ_{n} = e^{inx} be the nturn per second rotation (of radius 1) (sometimes called character). We want to write f as . We can prove (see mathematical derivation below) that the radii of the rotations (the coefficients c_{n}) are exactly the ones we gave in the previous paragraph.
For example, the plot of the function is closed, which means the function is periodic. The loop in the curve suggests that it is the sum of two periodic functions, one having a shorter period than the other. Indeed, it can be written: f(t) = e^{it} + e^{2it} = χ_{1}(t) + χ_{2}(t). All its Fourier coefficients are zero except c_{1} = 1 and c_{2} = 1. The graphical interpretation of a rotation is much harder to do than that of the translations because instead of visually seeing the movement from one point to another we have to add the whole motion for the decomposition to make sense (we are reasoning in rotation frequencies rather than in time).
Mathematically, adopting this point of view is seeing Fourier series as a tool to understand linear operators that commute with translations. The functions χ_{n} are precisely the multiplicative characters of the group .
Fourier series are named in honor of Joseph Fourier (17681830), who made important contributions to the study of trigonometric series, after preliminary investigations by Madhava, Nilakantha Somayaji, Jyesthadeva, Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. He applied this technique to find the solution of the heat equation, publishing his initial results in 1807 and 1811, and publishing his Théorie analytique de la chaleur in 1822.
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 (for example, one wondered if a function defined on two intervals with two different formulas was still a function). Later, Dirichlet and Riemann expressed Fourier's results with greater precision and formality.
In Fourier's work entitled Mémoire sur la propagation de la chaleur dans les corps solides, on pages 218 and 219, we can read the following :
 Multiplying both sides by , and then integrating from y = − 1 to y = + 1 yields:
In these few lines, which are surprisingly close to the modern formalism used in Fourier series, Fourier unwittingly revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli and Gauss, Fourier was the first to recognize that such trigonometric series could represent arbitrary functions, even those with discontinuities. It has required many years to clarify this insight, and it has led to important theories of convergence, function space, and harmonic analysis.
The originality of this work was such that when Fourier submitted his paper in 1807, the committee (composed of no lesser mathematicians than Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and that 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 mathematically equivalent (and correct), 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 to other applications the initial idea of representing any periodic function as a superposition of harmonics. This general area of inquiry is now sometimes called harmonic analysis.
The method used by Fourier to derive the coefficients of the series is very practical and wellsuited to the problem he was dealing with (heat propagation). However, this method has since been generalized to a much wider class of problems: writing a function as a sum of periodic functions.
More precisely, if f:R → C is a function, we would like to write this function as a sum of trigonometric functions, i.e. . We have to restrict our choice of functions in order for this to make sense. First of all, if f has period T, then by changing variables, can study which has period 2π. This simplifies notations a lot and allows us to use a canonical (standard) form. We can restrict the study of to any interval of length 2π, [π,π], say.
We will take the functions f:R → C in the set of piecewise continuous, 2π periodic functions with . Technically speaking, we are in fact taking functions from the Lp space L^{2}(μ), where μ is the normalized Lebesgue measure of the interval [π,π] (i.e. such that .
We can make L^{2}(μ) into a Hilbert space, which is wellsuited for orthogonal projections, by defining the scalar product:
where denotes the conjugate of f(x). We will denote by the associated norm.
is an orthonormal basis of L^{2}(μ), which means we can write
We usually define . These numbers are called complex Fourier coefficients. Their expression is
An equivalent formulation is to write f as a sum of sine and cosine functions.
The sum in the previous section is symmetrical around 0: indeed, except for n=0, a c_{n} coefficient corresponds to every c_{n} coefficient. This reminds one of the formulae
It is therefore possible to express the Fourier series with realvalued functions. To do this, we first notice that
After replacing c_{n} by its expression and simplifying the result we get
If, for a nonnegative integer n, we define the real Fourier coefficients a_{n} and b_{n} by
we get:
where f^{(k)} denotes the kth derivative of f.
This means that the sequence c_{n}(f) is rapidly decreasing.
Fourier series take advantage of the periodicity of a function f but what if f is periodic in more than one variable, or for that matter, what if f is not periodic? These problems led mathematicians and theoretical physicists to try to define Fourier series on any group G. The advantage of this is that it allows us, for example, to define Fourier series for functions of several variables. Fourier series and Fourier transforms usually used in signal processing then become special cases of this theory and are easier to interpret.
If G is a locally compact Abelian group and T is the unit circle, we can define the dual of G by . This is the set of rotations on the unit circle and its elements are called characters. We can define a scalar product on C[G] by: . is then an orthonormal basis of C[G] with respect to this scalar product. Let f :G → C. The Fourier coefficients of f are defined by: and we have . If the group is discrete, then the integral reduces to an ordinary sum.
For example, the Fourier series of this article are obtained by taking G=R/2πZ. We get
and
Periodic functions in n dimensions can be defined on an ndimensional torus (the function taking a value at each point on the torus). Such a torus is defined by T^{n}=R^{n}/(2π Z)^{n}. For n=1 we get a circle, for n=2 the cartesian product of two circles, i.e. a torus in the usual sense. Choosing G=T^{n} gives the corresponding Fourier series.
Let . We call Fourier series of the function f the series . For any positive integer N, we call the Nth partial sum of the Fourier series of this function.
Say we want to find the best approximation of f using only the functions χ_{n} for n from − N to N. Let . We are trying to find coefficients such that is minimum (where denotes the norm).
We have , where Re(z) denotes the real part of z.
Parseval's theorem (which can be derived independently from Fourier series) gives us
By definition, ; therefore
It is clear that this expression is minimum for x_{n} = c_{n} and for this value only.
This means that there is one and only one such that
it is given by
where
This means that the best approximation of f we can make using only the functions for n from − N to N is precisely the Nth partial sum of the Fourier series. An illustration of this is given on the animated plot of example 1.
While the Fourier coefficients a_{n} and b_{n} can be formally defined for any function for which the integrals make sense, whether the series so defined actually converges to f(x) depends on the properties of f.
The simplest answer is that if f is squareintegrable then
This is convergence in the norm of the space L^{2}. The proof of this result is simple, unlike Lennart Carleson's much stronger result that the series actually converges almost everywhere.
There are many known tests that ensure that the series converges at a given point x, for example, if the function is differentiable at x. Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at x, then the Fourier series will converge to the average of the left and right limits (but see Gibbs phenomenon). However, a fact that many find surprising, is that the Fourier series of a continuous function need not converge pointwise.
This unpleasant situation is counterbalanced by a theorem by Dirichlet which states that if f is 2Tperiodic and piecewise continuously differentiable function, then its Fourier series converges pointwise and , where and . If f is continuous as well as piecewise continuously differentiable, then the Fourier series converges uniformly.
In 1922, Andrey Kolmogorov published an article entitled Une série de FourierLebesgue divergente presque partout in which he gave an example of a Lebesgueintegrable function whose Fourier series diverges almost everywhere. This function is not in L^{2}(μ).
Another important property of the Fourier series is the Plancherel theorem. Let and c_{n}(f),c_{ n}(g) be the corresponding complex Fourier coefficients. Then
where denotes the conjugate of z.
Parseval's theorem, a special case of the Plancherel theorem, states that:
which can be restated with the real Fourier coefficients:
These theorems may be proven using the orthogonality relationships. They can be interpreted physically by saying that writing a signal as a Fourier series does not change its energy.
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Named after Joseph Fourier, a French mathematician
Singular 
Plural 
Fourier series (plural Fourier series)
