AFourier series (/ˈfʊrieɪ,-iər/[1]) is aseries expansion of aperiodic function into a sum oftrigonometric functions. The Fourier series is an example of atrigonometric series.[2] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used byJoseph Fourier to find solutions to theheat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not alwaysconverge. Well-behaved functions, for examplesmooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined byintegrals of the function multiplied by trigonometric functions, described inFourier series § Definition.
The study of theconvergence of Fourier series focus on the behaviors of thepartial sums, which means studying the behavior of the sum as more and more terms from the series are summed. The figures below illustrate some partial Fourier series results for the components of asquare wave.
A square wave (represented as the blue dot) is approximated by its sixth partial sum (represented as the purple dot), formed by summing the first six terms (represented as arrows) of the square wave's Fourier series. Each arrow starts at the vertical sum of all the arrows to its left (i.e. the previous partial sum).
The first four partial sums of the Fourier series for asquare wave. As more harmonics are added, the partial sumsconverge to (become more and more like) the square wave.
Function (in red) is a Fourier series sum of 6 harmonically related sine waves (in blue). Its Fourier transform is a frequency-domain representation that reveals the amplitudes of the summed sine waves.
Fourier series are closely related to theFourier transform, a more general tool that can even find the frequency information for functions that arenot periodic. Periodic functions can be identified with functions on a circle; for this reason Fourier series are the subject ofFourier analysis on thecircle group, denoted by or. The Fourier transform is also part ofFourier analysis, but is defined for functions on.
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 in Fourier's time. Fourier originally defined the Fourier series forreal-valued functions of real arguments, and used thesine and cosine functions in the decomposition. Many otherFourier-related transforms have since been defined, extending his initial idea to many applications and birthing anarea of mathematics calledFourier analysis.
The Fourier series is named in honor ofJean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study oftrigonometric series, after preliminary investigations byLeonhard Euler,Jean le Rond d'Alembert, andDaniel Bernoulli.[A] Fourier introduced the series for the purpose of solving theheat equation in a metal plate, publishing his initial results in his 1807Mémoire sur la propagation de la chaleur dans les corps solides (Treatise on the propagation of heat in solid bodies), and publishing hisThéorie analytique de la chaleur (Analytical theory of heat) in 1822. TheMémoire introduced Fourier analysis, specifically Fourier series. Through Fourier's research the fact was established that an arbitrary (at first, continuous[3] and later generalized to anypiecewise-smooth[4]) function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before theFrench Academy.[5] 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 ondeferents and epicycles.
Independently of Fourier, astronomerFriedrich Wilhelm Bessel introduced Fourier series to solveKepler's equation. His work was published in 1819, unaware of Fourier's work which remained unpublished until 1822.[6]
Theheat equation is apartial 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 asine orcosine wave. These simple solutions are now sometimes calledeigensolutions. Fourier's idea was to model a complicated heat source as a superposition (orlinear combination) of simple sine and cosine waves, and to write thesolution as a superposition of the correspondingeigensolutions. This superposition or linear combination is called the Fourier series.
This immediately gives any coefficientak of thetrigonometric 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 integralcan be carried out term-by-term. But all terms involving forj ≠k vanish when integrated from −1 to 1, leaving only the term, which is1.
In these few lines, which are close to the modernformalism used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used byEuler,d'Alembert,Daniel Bernoulli andGauss, 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 ofconvergence,function spaces, andharmonic analysis.
When Fourier submitted a later competition essay in 1811, the committee (which includedLagrange,Laplace,Malus andLegendre, 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 evenrigour".[14]
This resulting heat distribution in a metal plate is easily solved using Fourier's method
The Fourier series expansion of the sawtooth function (below) looks more complicated than the simple formula, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving theheat equation. For example, consider a metal plate in the shape of a square whose sides measure meters, with coordinates. 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, is maintained at the temperature gradient degrees Celsius, for in, then one can show that the stationary heat distribution (or the heat distribution after a long time has elapsed) is given by
Here, sinh is thehyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of the equation fromAnalysis § Example by. While our example function seems to have a needlessly complicated Fourier series, the heat distribution is nontrivial. The function cannot be written as aclosed-form expression. This method of solving the heat problem was made possible by Fourier's work.
Another application is to solve theBasel problem by usingParseval's theorem. The example generalizes and one may computeζ(2n), for any positive integern.
The Fourier series of a complex-valuedP-periodic function, integrable over the interval on the real line, is defined as atrigonometric series of the formsuch that theFourier coefficients are complex numbers defined by the integral[15][16]The series does not necessarily converge (in thepointwise sense) and, even if it does, it is not necessarily equal to. Only when certain conditions are satisfied (e.g. if is continuously differentiable) does the Fourier series converge to, i.e.,For functions satisfying theDirichlet sufficiency conditions, pointwise convergence holds.[17] However, these are notnecessary conditions and there are many theorems about different types ofconvergence of Fourier series (e.g.uniform convergence ormean convergence).[18] The definition naturally extends to the Fourier series of a (periodic)distribution (also calledFourier-Schwartz series).[19] Then the Fourier series converges to in the distribution sense.[20]
The process of determining the Fourier coefficients of a given function or signal is calledanalysis, while forming the associated trigonometric series (or its various approximations) is calledsynthesis.
A Fourier series can be written in several equivalent forms, shown here as thepartial sums of the Fourier series of:[21]
Fig 1. The top graph shows a non-periodic function in blue defined only over the red interval from 0 toP. The function can be analyzed over this interval to produce the Fourier series in the bottom graph. The Fourier series is always a periodic function, even if original function is not.
Sine-cosine form
Eq.1
Exponential form
Eq.2
The harmonics are indexed by an integer, which is also the number of cycles the corresponding sinusoids make in interval. Therefore, the sinusoids have:
These series can represent functions that are just a sum of one or more frequencies in theharmonic spectrum. In the limit, a trigonometric series can also represent the intermediate frequencies or non-sinusoidal functions because of the infinite number of terms.
The coefficients can be given/assumed, such as a music synthesizer or time samples of a waveform. In the latter case, the exponential form of Fourier series synthesizes adiscrete-time Fourier transform where variable represents frequency instead of time. In general, the coefficients are determined byanalysis of a given function whosedomain of definition is an interval of length.[B][22]
Plot of thesawtooth wave, a periodic continuation of the linear function on the intervalAnimated plot of the first five successive partial Fourier series
Consider a sawtooth function:In this case, the Fourier coefficients are given byIt can be shown that the Fourier series converges to at every point where is differentiable, and therefore:When, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of at. This is a particular instance of theDirichlet theorem for Fourier series.
This example leads to a solution of theBasel problem.
Fig 2. The blue curve is the cross-correlation of a square wave and a cosine template, as the phase lag of the template varies over one cycle. The amplitude and phase at the maximum value are the polar coordinates of one harmonic in the Fourier series expansion of the square wave. The corresponding rectangular coordinates can be determined by evaluating the correlation at just two samples separated by 90°.
An example of determining the parameter for one value of is shown in Figure 2. It is the value of at the maximum correlation between and a cosinetemplate,. The blue graph is thecross-correlation function, also known as amatched filter:
Fortunately, it is not necessary to evaluate this entire function, because its derivative is zero at the maximum: Hence
The notation is inadequate for discussing the Fourier coefficients of several different functions. Therefore, it is customarily replaced by a modified form of the function ( in this case), such as or and functional notation often replaces subscripting:
In engineering, particularly when the variable represents time, the coefficient sequence is called afrequency 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 tomodulate aDirac comb:
where represents a continuous frequency domain. When variable has units of seconds, has units ofhertz. The "teeth" of the comb are spaced at multiples (i.e.harmonics) of, which is called thefundamental frequency. can be recovered from this representation by aninverse Fourier transform:
The constructed function is therefore commonly referred to as aFourier transform, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.[C]
When the real and imaginary parts of a complex function are decomposed into theireven and odd parts, there are four components, denoted below by the subscriptsRE, 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:[29][30]
From this, various relationships are apparent, for example:
The transform of a real-valued function is theconjugate symmetric function Conversely, aconjugate symmetric transform implies a real-valued time-domain.
The transform of an imaginary-valued function is theconjugate antisymmetric function and the converse is true.
The transform of aconjugate symmetric function is the real-valued function and the converse is true.
The transform of aconjugate antisymmetric function is the imaginary-valued function and the converse is true.
If is a 2π-periodic function on which is times differentiable, and its derivative is continuous, then belongs to thefunction space.
If, then the Fourier coefficients of the derivative of can be expressed in terms of the Fourier coefficients of, via the formula In particular, since for any fixed we have as, it follows that tends to zero, i.e., the Fourier coefficients converge to zero faster than the power of.
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 anycompact group. Typical examples include thoseclassical groups that are compact. This generalizes the Fourier transform to all spaces of the formL2(G), whereG is a compact group, in such a way that the Fourier transform carriesconvolutions to pointwise products. The Fourier series exists and converges in similar ways to the[−π,π] case.
An alternative extension to compact groups is thePeter–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 is acompactRiemannian manifold, it has aLaplace–Beltrami operator. The Laplace–Beltrami operator is the differential operator that corresponds toLaplace operator for the Riemannian manifold. Then, by analogy, one can consider heat equations on. 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, where is a Riemannian manifold. The Fourier series converges in ways similar to the case. A typical example is to take to be the sphere with the usual metric, in which case the Fourier basis consists ofspherical harmonics.
This generalizes the Fourier transform to or, where is an LCA group. If is compact, one also obtains a Fourier series, which converges similarly to the case, but if is noncompact, one obtains instead aFourier integral. This generalization yields the usualFourier transform when the underlying locally compact Abelian group is.
Formally, the Fourier-Stieltjes series can be defined as the Fourier series whose coefficients are given byfor any, where is the spacefiniteBorel measures on the interval. As such, when, the function is also referred to as aFourier-Stieltjes transform.[32][33]
This follows from an earlier and more concrete representation of aRadon measure (i.e. alocally finiteBorel measure) on, given byF. Riesz. That is, if is function ofbounded variation on the interval then the Fourier coefficients can be expressed by theRiemann-Stieltjes integralcalled theFourier-Stieltjes coefficients of.[34] As the distributional derivative of is a Radon measure, it is subject to theLebesgue decomposition and can be expressed as.[35][36] If the expression reduces to the original definition of the Fourier coefficients, hence a Fourier series is a Fourier-Stieltjes series.
The Fourier series can be generalized still further from measures todistributions. If the Fourier coefficients are determined by a distribution then the series is sometimes described as aFourier-Schwartz series.[38]
While it is often extremely difficult to decide whether a given series is a Fourier or a Fourier-Stieltjes series, deciding whether or not it is a Fourier-Schwartz series is relatively trivial.[39]
We can also define the Fourier series for functions of two variables and 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 inimage compression. In particular, theJPEG image compression standard uses the two-dimensionaldiscrete cosine transform, a discrete form of theFourier cosine transform, which uses only cosine as the basis function.
For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry.[40]
Fourier series of a Bravais-lattice-periodic function
A three-dimensionalBravais lattice is defined as the set of vectors of the formwhere are integers and are three linearly independent but not necessarily orthogonal vectors. Let us consider some function with the same periodicity as the Bravais lattice,i.e. for any lattice vector. This situation frequently occurs insolid-state physics where might, for example, represent the effective potential that an electron "feels" inside a periodic crystal. In presence of such a periodic potential, the quantum-mechanical description of the electron results in a periodically modulated plane-wave commonly known asBloch state.
In order to develop in a Fourier series, it is convenient to introduce an auxiliary function Both and contain essentially the same information. However, instead of the position vector, the arguments of are coordinates along the unit vectors of the Bravais lattice, such that is an ordinary periodic function in these variables, This trick allows us to develop as a multi-dimensional Fourier series, in complete analogy with the square-periodic function discussed in the previous section. Its Fourier coefficients are where are all integers. plays the same role as the coefficients in the previous section but in order to avoid double subscripts we note them as a function.
Once we have these coefficients, the function can be recovered via the Fourier series We would now like to abandon the auxiliary coordinates and to return to the original position vector. This can be achieved by means of thereciprocal lattice whose vectors are defined such that they are orthonormal (up to a factor) to the original Bravais vectors,with theKronecker delta. With this, the scalar product between a reciprocal lattice vector and an arbitrary position vector written in the Bravais lattice basis becomeswhich is exactly the expression occurring in the Fourier exponents. The Fourier series for can therefore be rewritten as a sum over the all reciprocal lattice vectors, and the coefficients are The remaining task will be to convert this integral over lattice coordinates back into a volume integral. The relation between the lattice coordinates and the original cartesian coordinates is a linear system of equations,which, when written in matrix form,involves a constant matrix whose columns are the unit vectors of the Bravais lattice. When changing variables from to in an integral, the same matrix appears as aJacobian matrix
Its determinant is therefore also constant and can be inferred from any integral over any domain; here we choose to calculate the volume of the primitive unit cell in both coordinate systems: The unit cell being aparallelepiped, we have and thus This allows us to write as the desired volume integral over the primitive unit cell in ordinary cartesian coordinates:
Sines and cosines form an orthogonal set, as illustrated above. The integral of sine, cosine and their product is zero (green and red areas are equal, and cancel out) when, or the functions are different, and π only if and are equal, and the function used is the same. They would form an orthonormal set, if the integral equaled 1 (that is, each function would need to be scaled by).
The sine-cosine form follows in a similar fashion. Indeed, the sines and cosines form anorthogonal set:(whereδmn is theKronecker delta), andHence, the setalso forms an orthonormal basis for. The density of their span is a consequence of theStone–Weierstrass theorem, but follows also from the properties of classical kernels like theFejér kernel.
Fourier theorem proving convergence of Fourier series
Inengineering, the Fourier series is generally assumed to converge except at jump discontinuities since the functions encountered in engineering are usually better-behaved than those in other disciplines. In particular, if is continuous and the derivative of (which may not exist everywhere) is square integrable, then the Fourier series of converges absolutely and uniformly to.[42] If a function issquare-integrable on the interval, then the Fourier seriesconverges to the functionalmost everywhere. It is possible to define Fourier coefficients for more general functions or distributions, in which casepointwise convergence often fails, and convergence in norm orweak convergence is usually studied.
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)
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 convergence to a somewhat arbitrary function. Note the development of the "ringing" (Gibbs phenomenon) at the transitions to/from the vertical sections.
The theorems proving that a Fourier series is a valid representation of any periodic function (that satisfies theDirichlet conditions), and informal variations of them that do not specify the convergence conditions, are sometimes referred to generically asFourier's theorem orthe Fourier theorem.[43][44][45][46]
Theorem—The trigonometric polynomial is the unique best trigonometric polynomial of degree approximating, in the sense that, for any trigonometric polynomial of degree, 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 belongs to, then converges to in as, that is:
If is continuously differentiable, then is the Fourier coefficient of the first derivative. Since is continuous, and therefore bounded, it issquare-integrable and its Fourier coefficients are square-summable. Then, by theCauchy–Schwarz inequality,
This means that isabsolutely summable. The sum of this series is a continuous function, equal to, since the Fourier series converges in to:
This result can be proven easily if is further assumed to be, since in that case tends to zero as. More generally, the Fourier series is absolutely summable, thus converges uniformly to, provided that satisfies aHölder condition of order. 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 if is differentiable at, to more sophisticated results such asCarleson's theorem which states that the Fourier series of an function convergesalmost everywhere.
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 continuousT-periodic function need not converge pointwise. Theuniform boundedness principle yields a simple non-constructive proof of this fact.
In 1922,Andrey Kolmogorov published an article titledUne 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.[47]
It is possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic functionf defined for allx in [0,π] by[48]
Because the function is even the Fourier series contains only cosines:
The coefficients are:
Asm increases, the coefficients will be positive and increasing until they reach a value of about at for somen and then become negative (starting with a value around) and getting smaller, before starting a new such wave. At the Fourier series is simply the running sum of and this builds up to around
in thenth wave before returning to around zero, showing that the series does not converge at zero but reaches higher and higher peaks. Note that though the function is continuous, it is not differentiable.
Laurent series – the substitutionq = eix transforms a Fourier series into a Laurent series, or conversely. This is used in theq-series expansion of thej-invariant.
^Typically or. Some authors define because it simplifies the arguments of the sinusoid functions, at the expense of generality.
^Since the integral defining the Fourier transform of a periodic function is not convergent, it is necessary to view the periodic function and its transform asdistributions. In this sense is aDirac delta function, which is an example of a distribution.
^Fasshauer, Greg (2015)."Fourier Series and Boundary Value Problems"(PDF).Math 461 Course Notes, Ch 3. Department of Applied Mathematics, Illinois Institute of Technology. Retrieved6 November 2020.
^Wilhelm Flügge,Stresses in Shells (1973) 2nd edition.ISBN978-3-642-88291-3. Originally published in German asStatik und Dynamik der Schalen (1937).
^Fourier, Jean-Baptiste-Joseph (2014) [1890]. "Mémoire sur la propagation de la chaleur dans les corps solides, présenté le 21 Décembre 1807 à l'Institut national" [Report on the propagation of heat in solid bodies, presented on December 21, 1807 to the National Institute]. In Darboux, Gaston (ed.).Oeuvres de Fourier [The Works of Fourier] (in French). Vol. 2. Paris: Gauthier-Villars et Fils. pp. 218–219.doi:10.1017/CBO9781139568159.009.ISBN9781139568159. Whilst the cited article does list the author as Fourier, a footnote on page 215 indicates that the article was actually written byPoisson and that it is, "for reasons of historical interest", presented as though it were Fourier's original memoire.
^Kassam, Saleem A. (2004)."Fourier Series (Part II)"(PDF). Retrieved2024-12-11.The phase relationships are important because they correspond to having different amounts of "time shifts" or "delays" for each of the sinusoidal waveforms relative to a zero-phase waveform.
^abcdePapula, Lothar (2009).Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler [Mathematical Functions for Engineers and Physicists] (in German). Vieweg+Teubner Verlag.ISBN978-3834807571.
Boyce, William E.; DiPrima, Richard C. (2005).Elementary Differential Equations and Boundary Value Problems (8th ed.). New Jersey: John Wiley & Sons, Inc.ISBN0-471-43338-1.
Fourier, Joseph (2003).The Analytical Theory of Heat. Dover Publications.ISBN0-486-49531-0. 2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's workThéorie Analytique de la Chaleur, originally published in 1822.
Folland, Gerald B. (1992).Fourier analysis and its applications. Pacific Grove, Calif: Wadsworth & Brooks/Cole.ISBN978-0-534-17094-3.
Gonzalez-Velasco, Enrique A. (1992). "Connections in Mathematical Analysis: The Case of Fourier Series".American Mathematical Monthly.99 (5):427–441.doi:10.2307/2325087.JSTOR2325087.
