In mathematics, a Fourier series () is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.

History

The Fourier series is named in honor of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. 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 (at first, continuous and later generalized to any piecewise-smooth) 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. 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 and Bernhard Riemann 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, shell theory, etc.

Definition

Consider a real-valued function, $s(x)$, that is integrable on an interval of length $P$, which will be the period of the Fourier series. Common examples of analysis intervals are: :$x\; \backslash in,1$ and $P=1.$ :$x\; \backslash in\backslash pi,\backslash pi$ and $P=2\backslash pi.$ The analysis process determines the weights, indexed by integer $n$, which is also the number of cycles of the $n^\backslash text$ harmonic in the analysis interval. Therefore, the length of a cycle, in the units of $x$, is $P/n$. And the corresponding harmonic frequency is $n/P$. The $n^$ harmonics are $\backslash sin\backslash left(2\backslash pi\; x\; \backslash tfrac\backslash right)$ and $\backslash cos\backslash left(2\backslash pi\; x\; \backslash tfrac\backslash right)$, and their amplitudes (weights) are found by integration over the interval of length $P$: :*If $s(x)$ is $P$-periodic, then any interval of that length is sufficient. :*$a\_0$ and $b\_0$ can be reduced to $a\_0\; =\; \backslash frac\; \backslash int\_P\; s(x)\; \backslash ,\; dx$ and $b\_0\; =\; 0$. :*Many texts choose $P=2\backslash pi$ to simplify the argument of the sinusoid functions. The synthesis process (the actual Fourier series) is: In general, integer $N$ is theoretically infinite. Even so, the series might not converge or exactly equate to $s(x)$ at all values of $x$ (such as a single-point discontinuity) in the analysis interval. For the "well-behaved" functions typical of physical processes, equality is customarily assumed. Using a trigonometric identity: :$A\_n\backslash cdot\; \backslash cos\backslash left(\backslash tfracnx-\backslash varphi\_n\backslash right)\; \backslash \; \backslash equiv\; \backslash \; \backslash underbrace\_\backslash cdot\; \backslash cos\backslash left(\backslash tfracnx\backslash right)\; +\; \backslash underbrace\_\backslash cdot\; \backslash sin\backslash left(\backslash tfracnx\backslash right),$ and definitions $A\_n\; \backslash triangleq\; \backslash sqrt$ and $\backslash varphi\_n\; \backslash triangleq\; \backslash operatorname(b\_n,a\_n)$, the sine and cosine pairs can be expressed as a single sinusoid with a phase offset, analogous to the conversion between orthogonal (Cartesian) and polar coordinates: The customary form for generalizing to complex-valued $s(x)$ (next section) is obtained using Euler's formula to split the cosine function into complex exponentials. Here, complex conjugation is denoted by an asterisk: : Therefore, with definitions: :$c\_n\; \backslash triangleq\; \backslash left\backslash \backslash quad\; =\backslash quad\; \backslash frac\backslash int\_P\; s(x)\backslash cdot\; e^\backslash \; dx,$ the final result is:

Complex-valued functions

If $s(x)$ is a complex-valued function of a real variable $x,$ both components (real and imaginary part) are real-valued functions that can be represented by a Fourier series. The two sets of coefficients and the partial sum are given by: :$c\_\; =\; \backslash frac\backslash int\_P\; \backslash operatorname\backslash \backslash cdot\; e^\backslash \; dx$     and     $c\_\; =\; \backslash frac\backslash int\_P\; \backslash operatorname\backslash \backslash cdot\; e^\backslash \; dx$ :$s\_(x)\; =\; \backslash sum\_^N\; c\_\backslash cdot\; e^\; +\; i\backslash cdot\; \backslash sum\_^N\; c\_\backslash cdot\; e^\; =\backslash sum\_^N\; \backslash left(c\_+i\backslash cdot\; c\_\backslash right)\; \backslash cdot\; e^.$ Defining $c\_n\; \backslash triangleq\; c\_+i\backslash cdot\; c\_$ yields: This is identical to except $c\_n$ and $c\_$ are no longer complex conjugates. The formula for $c\_n$ is also unchanged: :

Other common notations

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

Convergence

In engineering applications, the Fourier series is generally presumed to converge almost everywhere (the exceptions being at discrete discontinuities) since the functions encountered in engineering are better-behaved than the functions that mathematicians can provide as counter-examples 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 $s(x)$. If a function is square-integrable on the interval $\_0,x\_0+P/math>,\; then\; the\; Fourier\; seriesconverges\; to\; the\; function\; at\; almost\; every\; point.\; Convergence\; of\; Fourier\; series\; also\; depends\; on\; the\; finite\; number\; of\; maxima\; and\; minima\; in\; a\; function\; which\; is\; popularly\; known\; as\; one\; of\; theDirichlet\text{'}s\; condition\; for\; Fourier\; series.\; SeeConvergence\; of\; Fourier\; series.\; It\; is\; possible\; to\; define\; Fourier\; coefficients\; for\; more\; general\; functions\; or\; distributions,\; in\; such\; cases\; convergence\; in\; norm\; orweak\; convergenceis\; usually\; of\; interest.Fourier\_series\_square\_wave\_circles\_animation.gif|link=|Four\; partial\; sums\; (Fourier\; series)\; of\; lengths\; 1,\; 2,\; 3,\; and\; 4\; terms,\; showing\; how\; the\; approximation\; to\; a\; square\; wave\; improves\; as\; the\; number\; of\; terms\; increases(animation)Fourier\_series\_sawtooth\_wave\_circles\_animation.gif|link=|Four\; partial\; sums\; (Fourier\; series)\; of\; lengths\; 1,\; 2,\; 3,\; and\; 4\; terms,\; showing\; how\; the\; approximation\; to\; a\; sawtooth\; wave\; improves\; as\; the\; number\; of\; terms\; increases(animation)Example\_of\_Fourier\_Convergence.gif\; |Example\; of\; convergence\; to\; a\; somewhat\; arbitrary\; function.\; Note\; the\; development\; of\; the\; "ringing"\; (Gibbs\; phenomenon)\; at\; the\; transitions\; to/from\; the\; vertical\; sections.An\; interactive\; animation\; can\; be\; see$
here.


Examples



Example 1: a simple Fourier series

thumb|right|400px|Animated plot of the first five successive partial Fourier series 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 $x=\backslash pi$, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of ''s'' at $x=\backslash pi$. This is a particular instance of the Dirichlet theorem for Fourier series. This example leads us to a solution to the Basel problem.

Example 2: Fourier's motivation

The Fourier series expansion of our function in Example 1 looks more complicated than the simple formula $s(x)=x/\; \backslash pi$, 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 sides measure $\backslash pi$ meters, with coordinates $(x,y)\; \backslash in,\backslash pi\backslash times,\backslash pi/math>.\; If\; there\; is\; no\; heat\; source\; within\; the\; plate,\; and\; if\; three\; of\; the\; four\; sides\; are\; held\; at\; 0\; degrees\; Celsius,\; while\; the\; fourth\; side,\; given\; by$ y=\backslash pi$,\; is\; maintained\; at\; the\; temperature\; gradient$ T(x,\backslash pi)=x$degrees\; Celsius,\; for$ x$in$ (0,\backslash pi)$,\; then\; one\; can\; show\; that\; the\; stationary\; heat\; distribution\; (or\; the\; heat\; distribution\; after\; a\; long\; period\; of\; time\; has\; elapsed)\; is\; given\; by\; :$ T(x,y)\; =\; 2\backslash sum\_^\backslash infty\; \backslash frac\; \backslash sin(nx)\; .$Here,\; sinh\; is\; thehyperbolic\; sinefunction.\; This\; solution\; of\; the\; heat\; equation\; is\; obtained\; by\; multiplying\; each\; term\; of\; \; by$ \backslash sinh(ny)/\backslash sinh(n\backslash pi)$.\; While\; our\; example\; function$ s(x)$seems\; to\; have\; a\; needlessly\; complicated\; Fourier\; series,\; the\; heat\; distribution$ T(x,y)$is\; nontrivial.\; The\; function$ T$cannot\; be\; written\; as\; aclosed-form\; expression.\; This\; method\; of\; solving\; the\; heat\; problem\; was\; made\; possible\; by\; Fourier\text{'}s\; work.$

Other applications

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(2''n''), for any positive integer ''n''.

Beginnings

Joseph Fourier wrote: This immediately gives any coefficient ''a_k'' of the trigonometrical series for φ(''y'') for any function which has such an expansion. It works because if φ has such an expansion, then (under suitable convergence assumptions) the integral : can be carried out term-by-term. But all terms involving $\backslash cos(2j+1)\backslash frac\; \backslash cos(2k+1)\backslash frac$ for vanish when integrated from −1 to 1, leaving only the ''k''th 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''.

Birth of harmonic analysis

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 real-valued functions of real arguments, and using the sine and cosine functions as the basis set for the decomposition. Many other Fourier-related 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.

Extensions




Fourier series on a square

We can also define the Fourier series for functions of two variables $x$ and $y$ in the square


Fourier series of Bravais-lattice-periodic-function

The three-dimensional Bravais lattice is defined as the set of vectors of the form: :$\backslash mathbf\; =\; n\_1\backslash mathbf\_1\; +\; n\_2\backslash mathbf\_2\; +\; n\_3\backslash mathbf\_3$ where $n\_i$ are integers and $\backslash mathbf\_i$ are three linearly independent vectors. Assuming we have some function, $f(\backslash mathbf)$, such that it obeys the following condition for any Bravais lattice vector $\backslash mathbf\; :\; f(\backslash mathbf)\; =\; f(\backslash mathbf+\backslash mathbf)$, 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 $\backslash mathbf$ in the coordinate-system of the lattice: : $\backslash mathbf\; =\; x\_1\backslash frac+\; x\_2\backslash frac+\; x\_3\backslash frac,$ where $a\_i\; \backslash triangleq\; |\backslash mathbf\_i|.$ Thus we can define a new function, : $g(x\_1,x\_2,x\_3)\; \backslash triangleq\; f(\backslash mathbf)\; =\; f\; \backslash left\; (x\_1\backslash frac+x\_2\backslash frac+x\_3\backslash frac\; \backslash right\; ).$ This new function, $g(x\_1,x\_2,x\_3)$, is now a function of three-variables, each of which has periodicity ''a''₁, ''a''₂, ''a''₃ respectively: : $g(x\_1,x\_2,x\_3)\; =\; g(x\_1+a\_1,x\_2,x\_3)\; =\; g(x\_1,x\_2+a\_2,x\_3)\; =\; g(x\_1,x\_2,x\_3+a\_3).$ This enables us to build up a set of Fourier coefficients, each being indexed by three independent integers $m\_1,m\_2,m\_3$. In what follows, we use function notation to denote these coefficents, where previously we used subscripts. If we write a series for ''g'' on the interval , ''a''₁for ''x''₁, we can define the following: :$h^\backslash mathrm(m\_1,\; x\_2,\; x\_3)\; \backslash triangleq\; \backslash frac\backslash int\_0^\; g(x\_1,\; x\_2,\; x\_3)\backslash cdot\; e^\backslash ,\; dx\_1$ And then we can write: :$g(x\_1,\; x\_2,\; x\_3)=\backslash sum\_^\backslash infty\; h^\backslash mathrm(m\_1,\; x\_2,\; x\_3)\; \backslash cdot\; e^$ Further defining: : We can write $g$ once again as: :$g(x\_1,\; x\_2,\; x\_3)=\backslash sum\_^\backslash infty\; \backslash sum\_^\backslash infty\; h^\backslash mathrm(m\_1,\; m\_2,\; x\_3)\; \backslash cdot\; e^\; \backslash cdot\; e^$ Finally applying the same for the third coordinate, we define: : We write $g$ as: :$g(x\_1,\; x\_2,\; x\_3)=\backslash sum\_^\backslash infty\; \backslash sum\_^\backslash infty\; \backslash sum\_^\backslash infty\; h^\backslash mathrm(m\_1,\; m\_2,\; m\_3)\; \backslash cdot\; e^\; \backslash cdot\; e^\backslash cdot\; e^$ Re-arranging: :$g(x\_1,\; x\_2,\; x\_3)=\backslash sum\_\; h^\backslash mathrm(m\_1,\; m\_2,\; m\_3)\; \backslash cdot\; e^.$ Now, every ''reciprocal'' lattice vector can be written as $\backslash mathbf\; =\; \backslash ell\_1\backslash mathbf\_1\; +\; \backslash ell\_2\backslash mathbf\_2\; +\; \backslash ell\_3\backslash mathbf\_3$, where $l\_i$ are integers and $\backslash mathbf\_i$ are the reciprocal lattice vectors, we can use the fact that $\backslash mathbf\; \backslash cdot\; \backslash mathbf=2\backslash pi\backslash delta\_$ to calculate that for any arbitrary reciprocal lattice vector $\backslash mathbf$ and arbitrary vector in space $\backslash mathbf$, their scalar product is: :$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; \backslash left\; (\; \backslash ell\_1\backslash mathbf\_1\; +\; \backslash ell\_2\backslash mathbf\_2\; +\; \backslash ell\_3\backslash mathbf\_3\; \backslash right\; )\; \backslash cdot\; \backslash left\; (x\_1\backslash frac+\; x\_2\backslash frac\; +x\_3\backslash frac\; \backslash right\; )\; =\; 2\backslash pi\; \backslash left(\; x\_1\backslash frac+x\_2\backslash frac+x\_3\backslash frac\; \backslash right\; ).$ And so it is clear that in our expansion, the sum is actually over reciprocal lattice vectors: :$f(\backslash mathbf)=\backslash sum\_\; h(\backslash mathbf)\; \backslash cdot\; e^,$ where : $h(\backslash mathbf)\; =\; \backslash frac\; \backslash int\_0^\; dx\_3\; \backslash ,\; \backslash frac\backslash int\_0^\; dx\_2\; \backslash ,\; \backslash frac\backslash int\_0^\; dx\_1\; \backslash ,\; f\backslash left(x\_1\backslash frac\; +\; x\_2\backslash frac\; +\; x\_3\backslash frac\; \backslash right)\backslash cdot\; e^.$ Assuming :$\backslash mathbf\; =\; (x,y,z)\; =\; x\_1\backslash frac+x\_2\backslash frac+x\_3\backslash frac,$ 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 non-trivial cross-product identities can be shown to be equal to: : $\backslash frac$ (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 $\backslash mathbf\_1$ is parallel to the ''x'' axis, $\backslash mathbf\_2$ lies in the ''xy''-plane, and $\backslash mathbf\_3$ has components of all three axes). The denominator is exactly the volume of the primitive unit cell which is enclosed by the three primitive-vectors $\backslash mathbf\_1$, $\backslash mathbf\_2$ and $\backslash mathbf\_3$. In particular, we now know that :$dx\_1\; \backslash ,\; dx\_2\; \backslash ,\; dx\_3\; =\; \backslash frac\; \backslash cdot\; dx\; \backslash ,\; dy\; \backslash ,\; dz.$ We can write now $h(\backslash mathbf)$ 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: :$h(\backslash mathbf)\; =\; \backslash frac\backslash int\_\; d\backslash mathbf\; f(\backslash mathbf)\backslash cdot\; e^$ writing $d\backslash mathbf$ for the volume element $dx\; \backslash ,\; dy\; \backslash ,\; dz$; and where $C$ is the primitive unit cell, thus, $\backslash mathbf\_1\backslash cdot(\backslash mathbf\_2\; \backslash times\; \backslash mathbf\_3)$ is the volume of the primitive unit cell.


Hilbert space interpretation

In the language of Hilbert spaces, the set of functions $\backslash left\backslash$ is an orthonormal basis for the space $L^2(\backslash pi,\backslash pi$ of square-integrable functions on $\backslash pi,\backslash pi/math>.\; This\; space\; is\; actually\; a\; Hilbert\; space\; with\; aninner\; productgiven\; for\; any\; two\; elements$ f$and$ g$by:\; :$ \backslash langle\; f,\backslash ,\; g\; \backslash rangle\; \backslash ;\backslash triangleq\; \backslash ;\; \backslash frac\backslash int\_^\; f(x)g^*(x)\backslash ,dx,$ \; \; where$ g^(x)$is\; the\; complex\; conjugate\; of$ g(x).$The\; basic\; Fourier\; series\; result\; for\; Hilbert\; spaces\; can\; be\; written\; as\; :$ f=\backslash sum\_^\backslash infty\; \backslash langle\; f,e\_n\; \backslash rangle\; \backslash ,\; e\_n.$$ 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: :$\backslash int\_^\; \backslash cos(mx)\backslash ,\; \backslash cos(nx)\backslash ,\; dx\; =\; \backslash frac\backslash int\_^\; \backslash cos((n-m)x)+\backslash cos((n+m)x)\backslash ,\; dx\; =\; \backslash pi\; \backslash delta\_,\; \backslash quad\; m,\; n\; \backslash ge\; 1,$ :$\backslash int\_^\; \backslash sin(mx)\backslash ,\; \backslash sin(nx)\backslash ,\; dx\; =\; \backslash frac\backslash int\_^\; \backslash cos((n-m)x)-\backslash cos((n+m)x)\backslash ,\; dx\; =\; \backslash pi\; \backslash delta\_,\; \backslash quad\; m,\; n\; \backslash ge\; 1$ (where ''δ''_''mn'' is the Kronecker delta), and :$\backslash int\_^\; \backslash cos(mx)\backslash ,\; \backslash sin(nx)\backslash ,\; dx\; =\; \backslash frac\backslash int\_^\; \backslash sin((n+m)x)+\backslash sin((n-m)x)\backslash ,\; dx\; =\; 0;\backslash ,$ furthermore, the sines and cosines are orthogonal to the constant function $1$. An ''orthonormal basis'' for $L^2(\backslash pi,\backslash pi$ consisting of real functions is formed by the functions $1$ and $\backslash sqrt\; \backslash cos\; (nx)$, $\backslash sqrt\; \backslash sin\; (nx)$ with ''n'' = 1, 2, â€¦. 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.


Properties



Table of basic properties

This table shows some mathematical operations in the time domain and the corresponding effect in the Fourier series coefficients. Notation: * Complex conjugation is denoted by an asterisk. *$f(x),g(x)$ designate $P$-periodic functions or functions defined only for $x\; \backslash in,P$. * $FG/math>\; designate\; the\; Fourier\; series\; coefficients\; (exponential\; form)\; of$ f$and$ g$as\; defined\; in\; equation\; .$


Symmetry properties

When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform: : From this, various relationships are apparent, for example: *The transform of a real-valued function () is the even symmetric function . Conversely, an even-symmetric transform implies a real-valued time-domain. *The transform of an imaginary-valued function () is the odd symmetric function , and the converse is true. *The transform of an even-symmetric function () is the real-valued function , and the converse is true. *The transform of an odd-symmetric function () is the imaginary-valued function , and the converse is true.


Riemann–Lebesgue lemma

If $f$ is integrable, $\backslash lim\_F0$, $\backslash lim\_a\_n=0$ and $\backslash lim\_b\_n=0.$ This result is known as the Riemann–Lebesgue lemma.


Parseval's theorem

If $f$ belongs to $L^2(P)$ (an interval of length $P$) then: $\backslash sum\_^\backslash infty\; \backslash bigg|Fbigg|^2\; =\; \backslash frac\backslash int\_\; |f(x)|^2\; \backslash ,\; dx.$


Plancherel's theorem

If $c\_0,\backslash ,\; c\_,\backslash ,\; c\_,\backslash ldots$ are coefficients and then there is a unique function $f\backslash in\; L^2(P)$ such that $F=\; c\_n$ for every $n$.


Convolution theorems

Given $P$-periodic functions, $f\_$ and $g\_$ with Fourier series coefficients $F/math>\; and$ G$$ n\; \backslash in\; \backslash mathbb,$*The\; pointwise\; product:\; $ h\_(x)\; \backslash triangleq\; f\_(x)\backslash cdot\; g\_(x)$is\; also$ P$-periodic,\; and\; its\; Fourier\; series\; coefficients\; are\; given\; by\; thediscrete\; convolutionof\; the$ F$and$ G$sequences:\; :$ H=\; \backslash $*Theperiodic\; convolution:\; $ h\_(x)\; \backslash triangleq\; \backslash int\_\; f\_(\backslash tau)\backslash cdot\; g\_(x-\backslash tau)\backslash ,\; d\backslash tau$is\; also$ P$-periodic,\; with\; Fourier\; series\; coefficients:\; :$ H=\; P\; \backslash cdot\; Fcdot\; G$*Adoubly\; infinitesequence$ \backslash left\; \backslash \_$in$ c\_0(\backslash mathbb)$is\; the\; sequence\; of\; Fourier\; coefficients\; of\; a\; function\; in$ L^1(,2\backslash pi$if\; and\; only\; if\; it\; is\; a\; convolution\; of\; two\; sequences\; in$ \backslash ell^2(\backslash mathbb)$.\; See$


Derivative property

We say that $f$ belongs to $C^k(\backslash mathbb)$ if $f$ is a 2-periodic function on $\backslash mathbb$ which is $k$ times differentiable, and its ''k''^th derivative is continuous. * If $f\; \backslash in\; C^1(\backslash mathbb)$, then the Fourier coefficients $\backslash widehat/math>\; of\; the\; derivative$ f\text{'}$can\; be\; expressed\; in\; terms\; of\; the\; Fourier\; coefficients$ \backslash widehat/math>\; of\; the\; function$ f$,\; via\; the\; formula$ \backslash widehat=\; in\; \backslash widehat/math>.\; *\; If$ f\; \backslash in\; C^k(\backslash mathbb)$,\; then$ \backslash widehat=\; (in)^k\; \backslash widehat/math>.\; In\; particular,\; since\; for\; a\; fixed$ k\backslash geq\; 1$we\; have$ \backslash widehatto\; 0$as$ n\backslash to\backslash infty$,\; it\; follows\; that$ |n|^k\backslash widehat/math>\; tends\; to\; zero,\; which\; means\; that\; the\; Fourier\; coefficients\; converge\; to\; zero\; faster\; than\; the\; \text{'}\text{'}k\text{'}\text{'}th\; power\; of\; \text{'}\text{'}n\text{'}\text{'}\; for\; any$ k\backslash geq\; 1$.$$$$$


Compact groups

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 ˆ’,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.


Riemannian manifolds

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 Laplace–Beltrami operator. The Laplace–Beltrami 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 Laplace–Beltrami 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 $\backslash pi,\backslash pi/math>\; case.\; A\; typical\; example\; is\; to\; take$ X$to\; be\; the\; sphere\; with\; the\; usual\; metric,\; in\; which\; case\; the\; Fourier\; basis\; consists\; ofspherical\; harmonics.$


Locally compact Abelian groups

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 $\backslash pi,\backslash pi/math>\; case,\; but\; if$ G$is\; noncompact,\; one\; obtains\; instead\; aFourier\; integral.\; This\; generalization\; yields\; the\; usualFourier\; transformwhen\; the\; underlying\; locally\; compact\; Abelian\; group\; is$ \backslash mathbb$.$


Table of common Fourier series

Some common pairs of periodic functions and their Fourier Series coefficients are shown in the table below. The following notation applies: *$f(x)$ designates a periodic function defined on . *$a\_0,\; a\_n,\; b\_n$ designate the Fourier Series coefficients (sine-cosine form) of the periodic function $f$ as defined in .

Approximation and convergence of Fourier series

Recalling , :$s\_(x)\; =\; \backslash sum\_^N\; Se^,$ it is a trigonometric polynomial of degree $N$, generally: :$p\_(x)=\backslash sum\_^N\; pe^.$

Least squares property

Parseval's theorem implies that:

Theorem. The trigonometric polynomial $s\_$ is the unique best trigonometric polynomial of degree $N$ approximating $s(x)$, in the sense that, for any trigonometric polynomial $p\_\; \backslash neq\; s\_$ of degree $N$, we have: : where the Hilbert space norm is defined as: :$\backslash |\; g\; \backslash |\_2\; =\; \backslash sqrt.$

Convergence

Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result. Theorem. If $s$ belongs to $L^2\; (P)$ (an interval of length $P$), then $s\_$ converges to $s$ in $L^2\; (P)$, that is,  $\backslash |s\_\; -\; s\backslash |\_2$ converges to 0 as $N\; \backslash rightarrow\; \backslash infty$. We have already mentioned that if $s$ is continuously differentiable, then  $(i\backslash cdot\; n)\; S/math> \; is\; the\; \text{'}\text{'}n\text{'}\text{'}thFourier\; coefficient\; of\; the\; derivative$ s\text{'}$.\; It\; follows,\; essentially\; from\; theCauchy–Schwarz\; inequality,\; that$ s\_$is\; absolutely\; summable.\; The\; sum\; of\; this\; series\; is\; a\; continuous\; function,\; equal\; to$ s$,\; since\; the\; Fourier\; series\; converges\; in\; the\; mean\; to$ s$:\; Theorem.\; If$ s\; \backslash in\; C^1(\backslash mathbb)$,\; then$ s\_$converges\; to$ s$uniformly(and\; hence\; alsopointwise.)\; This\; result\; can\; be\; proven\; easily\; if$ s$is\; further\; assumed\; to\; be$ C^2$,\; since\; in\; that\; case$ n^2S/math>\; tends\; to\; zero\; as$ n\; \backslash rightarrow\; \backslash infty$.\; More\; generally,\; the\; Fourier\; series\; is\; absolutely\; summable,\; thus\; converges\; uniformly\; to$ s$,\; provided\; that$ s$satisfies\; aHÃ\P lder\; conditionof\; order$ \backslash alpha\; >\; 1/2$.\; In\; the\; absolutely\; summable\; case,\; the\; inequality:\; :$ \backslash sup\_x\; |s(x)\; -\; s\_(x)|\; \backslash le\; \backslash sum\_\; |S$ \; proves\; uniform\; convergence.\; Many\; other\; results\; concerning\; theconvergence\; of\; Fourier\; seriesare\; known,\; ranging\; from\; the\; moderately\; simple\; result\; that\; the\; series\; converges\; at$ x$if$ s$is\; differentiable\; at$ x$,\; toLennart\; Carleson\text{'}s\; much\; more\; sophisticated\; result\; that\; the\; Fourier\; series\; of\; an$ L^2$function\; actually\; convergesalmost\; everywhere.\; These\; theorems,\; and\; informal\; variations\; of\; them\; that\; don\text{'}t\; specify\; the\; convergence\; conditions,\; are\; sometimes\; referred\; to\; generically\; as\; "Fourier\text{'}s\; theorem"\; or\; "the\; Fourier\; theorem".$$


Divergence

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 ''T''-periodic function need not converge pointwise. The uniform boundedness principle yields a simple non-constructive proof of this fact. In 1922, Andrey Kolmogorov published an article titled ''Une série de Fourier-Lebesgue divergente presque partout'' in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. He later constructed an example of an integrable function whose Fourier series diverges everywhere .

See also

* ATS theorem * Dirichlet kernel * Discrete Fourier transform * Fast Fourier transform * Fejér's theorem * Fourier analysis * Fourier sine and cosine series * Fourier transform * Gibbs phenomenon * Laurent series – the substitution ''q'' = ''e''^''ix'' transforms a Fourier series into a Laurent series, or conversely. This is used in the ''q''-series expansion of the ''j''-invariant. * Least-squares spectral analysis * Multidimensional transform * Spectral theory * Sturm–Liouville theory * Residue Theorem integrals of f singularities, poles


Notes



References



Further reading

* * 2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's work ''Théorie Analytique de la Chaleur'', originally published in 1822. * * * * Felix Klein, ''Development of mathematics in the 19th century''. Mathsci Press Brookline, Mass, 1979. Translated by M. Ackerman from ''Vorlesungen über die Entwicklung der Mathematik im 19 Jahrhundert'', Springer, Berlin, 1928. * * The first edition was published in 1935.

External links

* * * * {{DEFAULTSORT:Fourier Series Category:Joseph Fourier