In mathematics, a Fourier series (/ˈfʊrieɪ,-iər/) 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.
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.
Function (in red) is a sum of six sine functions of different amplitudes and harmonically related frequencies. Their summation is called a Fourier series. The Fourier transform, (in blue), which depicts amplitude vs frequency, reveals the 6 frequencies (at odd harmonics) and their amplitudes (1/odd number).
The first four partial sums of the Fourier series for a square wave
Consider a real-valued function, that is integrable on an interval of length which will be the period of the Fourier series. The correlation function:
is essentially a matched filter, with template Its peak value is a relative measure of the presence of frequency in function The analysis process determines, for certain key frequencies, the maximum correlation and the corresponding phase offset, The synthesis process (the actual Fourier series), in terms of parameters to be determined by analysis, is:
Fourier series, amplitude-phase form
In general, integer is theoretically infinite. Even so, the series might not converge or exactly equate to at all values of (such as a single-point discontinuity) in the analysis interval. For the "well-behaved" functions typical of physical processes, equality is customarily assumed.
Integer used as an index, is also the number of cycles of the -th harmonic in interval Therefore, the length of a cycle, in the units of is The corresponding harmonic frequency is so the -th harmonic is Some texts define to simplify the argument of the sinusoid functions at the expense of generality.
Clearly Eq.1 can represent functions that are just a sum of one or more of the harmonic frequencies. The remarkable thing, for those not yet familiar with this concept, is that it can also represent the intermediate frequencies and/or non-sinusoidal functions because of the potentially infinite number of terms ().
The sum of the sinusoidal components of a Fourier series is a periodic function, whether or not the original function, s(x), is periodic.
If is a function contained in an interval of length (and zero elsewhere), the upper-right quadrant is an example of what its Fourier series coefficients () might look like when plotted against their corresponding harmonic frequencies. The upper-left quadrant is the corresponding Fourier transform of The Fourier series summation (not shown) synthesizes a periodic summation of whereas the inverse Fourier transform (not shown) synthesizes only
Rather than computationally intensive cross-correlation, Fourier analysis customarily exploits a trigonometric identity:
where parameters and replace and and can be found by evaluating the cross-correlation at only two values of phase:
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 is continuous and the derivative of (which may not exist everywhere) is square integrable, then the Fourier series of converges absolutely and uniformly to . If a function is square-integrable on the interval , then the Fourier series converges to the function at almost every point. Convergence of Fourier series also depends on the finite number of maxima and minima in a function which is popularly known as one of the Dirichlet's condition for Fourier series. See Convergence of Fourier series. It is possible to define Fourier coefficients for more general functions or distributions, in such cases convergence in norm or weak convergence is usually of interest.
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.
If is a complex-valued function of a real variable 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:
This is identical to Eq.4 except and are no longer complex conjugates. The formula for is also unchanged:
Other common notations
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 a frequency domain representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.
Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:
The constructed function is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.[A]
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.[B] 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.
This immediately gives any coefficient ak 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 for j ≠ k vanish when integrated from −1 to 1, leaving only the kth term.
In these few lines, which are close to the modern formalism used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli and Gauss, Fourier believed that such trigonometric series could represent any arbitrary function. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis.
When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.
Heat distribution in a metal plate, using Fourier's method
The Fourier series expansion of the sawtooth function (above) 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 the heat 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 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.5 by . While our example function seems to have a needlessly complicated Fourier series, the heat distribution is nontrivial. The function cannot be written as a closed-form expression. This method of solving the heat problem was made possible by Fourier's work.
Complex Fourier series tracing the letter 'e'. (The Julia source code that generates the frames of this animation is here in Appendix B.)
An example of the ability of the complex Fourier series to trace any two dimensional closed figure is shown in the adjacent animation of the complex Fourier series tracing the letter 'e' (for exponential). Note that the animation uses the variable 't' to parameterize the letter 'e' in the complex plane, which is equivalent to using the parameter 'x' in this article's subsection on complex valued functions.
In the animation's back plane, the rotating vectors are aggregated in an order that alternates between a vector rotating in the positive (counter clockwise) direction and a vector rotating at the same frequency but in the negative (clockwise) direction, resulting in a single tracing arm with lots of zigzags. This perspective shows how the addition of each pair of rotating vectors (one rotating in the positive direction and one rotating in the negative direction) nudges the previous trace (shown as a light gray dotted line) closer to the shape of the letter 'e'.
In the animation's front plane, the rotating vectors are aggregated into two sets, the set of all the positive rotating vectors and the set of all the negative rotating vectors (the non-rotating component is evenly split between the two), resulting in two tracing arms rotating in opposite directions. The animation's small circle denotes the midpoint between the two arms and also the midpoint between the origin and the current tracing point denoted by '+'. This perspective shows how the complex Fourier series is an extension (the addition of an arm) of the complex geometric series which has just one arm. It also shows how the two arms coordinate with each other. For example, as the tracing point is rotating in the positive direction, the negative direction arm stays parked. Similarly, when the tracing point is rotating in the negative direction, the positive direction arm stays parked.
In between the animation's back and front planes are rotating trapezoids whose areas represent the values of the complex Fourier series terms. This perspective shows the amplitude, frequency, and phase of the individual terms of the complex Fourier series in relation to the series sum spatially converging to the letter 'e' in the back and front planes. The audio track's left and right channels correspond respectively to the real and imaginary components of the current tracing point '+' but increased in frequency by a factor of 3536 so that the animation's fundamental frequency (n=1) is a 220 Hz tone (A220).
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.
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:
designates a periodic function defined on .
designate the Fourier Series coefficients (sine-cosine form) of the periodic function as defined in Eq.2.
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 (fRE+ fRO) is the even symmetric function FRE+ i FIO. Conversely, an even-symmetric transform implies a real-valued time-domain.
The transform of an imaginary-valued function (ifIE+ ifIO) is the odd symmetric function FRO+ i FIE, and the converse is true.
The transform of an even-symmetric function (fRE+ ifIO) is the real-valued function FRE+ FRO, and the converse is true.
The transform of an odd-symmetric function (fRO+ ifIE) is the imaginary-valued function i FIE+ i FIO, and the converse is true.
is also -periodic, with Fourier series coefficients:
A doubly infinite sequence in is the sequence of Fourier coefficients of a function in if and only if it is a convolution of two sequences in . See 
We say that belongs to
if is a 2π-periodic function on which is times differentiable, and its kth derivative is continuous.
If , then the Fourier coefficients of the derivative can be expressed in terms of the Fourier coefficients of the function , via the formula .
If , then . In particular, since for a fixed we have as , it follows that tends to zero, which means that the Fourier coefficients converge to zero faster than the kth power of n for any .
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 L2(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 is a compactRiemannian manifold, it has a Laplace–Beltrami operator. The Laplace–Beltrami operator is the differential operator that corresponds to Laplace 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 of spherical 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 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 a Fourier integral. This generalization yields the usual Fourier transform when the underlying locally compact Abelian group is .
Fourier series on a square
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 in image compression. In particular, the jpeg image compression standard uses the two-dimensional discrete cosine transform, which is a Fourier-related transform using only the cosine basis functions.
For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry.
Fourier series of Bravais-lattice-periodic-function
A three-dimensional Bravais lattice is defined as the set of vectors of the form:
where are integers and are three linearly independent vectors. Assuming we have some function, , such that it obeys the condition of periodicity for any Bravais lattice vector , , 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 the Fourier series of the potential when applying Bloch's theorem. First, we may write any arbitrary position vector in the coordinate-system of the lattice:
where meaning that is defined to be the magnitude of , so is the unit vector directed along .
Thus we can define a new function,
This new function, , is now a function of three-variables, each of which has periodicity , , and respectively:
This enables us to build up a set of Fourier coefficients, each being indexed by three independent integers . In what follows, we use function notation to denote these coefficients, where previously we used subscripts. If we write a series for on the interval for , we can define the following:
And then we can write:
We can write once again as:
Finally applying the same for the third coordinate, we define:
We write as:
Now, every reciprocal lattice vector can be written (but does not mean that it is the only way of writing) as , where are integers and are reciprocal lattice vectors to satisfy ( for , and for ). Then for any arbitrary reciprocal lattice vector and arbitrary position vector in the original Bravais lattice space, their scalar product is:
So it is clear that in our expansion of , the sum is actually over reciprocal lattice vectors:
we can solve this system of three linear equations for , , and in terms of , and in order to calculate the volume element in the original cartesian coordinate system. Once we have , , and in terms of , and , 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:
(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 is parallel to the x axis, lies in the xy-plane, and 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 , and . In particular, we now know that
We can write now as an integral with the traditional coordinate system over the volume of the primitive cell, instead of with the , and variables:
writing for the volume element ; and where is the primitive unit cell, thus, is the volume of the primitive unit cell.
Hilbert space interpretation
In the language of Hilbert spaces, the set of functions is an orthonormal basis for the space of square-integrable functions on . This space is actually a Hilbert space with an inner product given for any two elements and by:
where is the complex conjugate of
The basic Fourier series result for Hilbert spaces can be written as
Sines and cosines form an orthonormal 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.
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:
furthermore, the sines and cosines are orthogonal to the constant function . An orthonormal basis for consisting of real functions is formed by the functions and , with 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.
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 (an interval of length ), then converges to in , that is, converges to 0 as .
We have already mentioned that if is continuously differentiable, then is the nth Fourier coefficient of the derivative . It follows, essentially from the Cauchy–Schwarz inequality, that is absolutely summable. The sum of this series is a continuous function, equal to , since the Fourier series converges in the mean 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 a Hölder condition of order . In the absolutely summable case, the inequality:
These theorems, and informal variations of them that don't specify the convergence conditions, are sometimes referred to generically as "Fourier's theorem" or "the Fourier theorem".
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 (Katznelson 1976).
^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 as distributions. In this sense is a Dirac delta function, which is an example of a distribution.
^These words are not strictly Fourier's. Whilst the cited article does list the author as Fourier, a footnote indicates that the article was actually written by Poisson (that it was not written by Fourier is also clear from the consistent use of the third person to refer to him) and that it is, "for reasons of historical interest", presented as though it were Fourier's original memoire.
^Dorf, Richard C.; Tallarida, Ronald J. (1993). Pocket Book of Electrical Engineering Formulas (1st ed.). Boca Raton,FL: CRC Press. pp. 171–174. ISBN 0849344735.
^Tolstov, Georgi P. (1976). Fourier Series. Courier-Dover. ISBN 0-486-63317-9.
^Wolfram, Eric W. "Fourier Series (eq.30)". MathWorld--A Wolfram Web Resource. Retrieved 3 November 2021.
^Cheever, Erik. "Derivation of Fourier Series". lpsa.swarthmore.edu. Retrieved 3 November 2021.
^Stillwell, John (2013). "Logic and the philosophy of mathematics in the nineteenth century". In Ten, C. L. (ed.). Routledge History of Philosophy. Volume VII: The Nineteenth Century. Routledge. p. 204. ISBN 978-1-134-92880-4. |volume= has extra text (help)
^Fasshauer, Greg (2015). "Fourier Series and Boundary Value Problems" (PDF). Math 461 Course Notes, Ch 3. Department of Applied Mathematics, Illinois Institute of Technology. Retrieved 6 November 2020.
^Mascre, D.; Riemann, Bernhard (1867), "Posthumous Thesis on the Representation of Functions by Trigonometric Series", in Grattan-Guinness, Ivor (ed.), Landmark Writings in Western Mathematics 1640–1940, Elsevier (published 2005), p. 49, ISBN 9780080457444
^Remmert, Reinhold (1991). Theory of Complex Functions: Readings in Mathematics. Springer. p. 29. ISBN 9780387971957.
^Nerlove, Marc; Grether, David M.; Carvalho, Jose L. (1995). Analysis of Economic Time Series. Economic Theory, Econometrics, and Mathematical Economics. Elsevier. ISBN 0-12-515751-7.
^Wilhelm Flügge, Stresses in Shells (1973) 2nd edition. ISBN 978-3-642-88291-3. Originally published in German as Statik und Dynamik der Schalen (1937).
^Fourier, Jean-Baptiste-Joseph (1888). Gaston Darboux (ed.). Oeuvres de Fourier [The Works of Fourier] (in French). Paris: Gauthier-Villars et Fils. pp. 218–219 – via Gallica.
^Sepesi, G. "Zeno's Enduring Example". Towards Data Science. pp. Appendix B.
^ abcdePapula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler [Mathematical Functions for Engineers and Physicists] (in German). Vieweg+Teubner Verlag. ISBN 978-3834807571.
^ abcdShmaliy, Y.S. (2007). Continuous-Time Signals. Springer. ISBN 978-1402062711.
^Proakis, John G.; Manolakis, Dimitris G. (1996). Digital Signal Processing: Principles, Algorithms, and Applications (3rd ed.). Prentice Hall. p. 291. ISBN 978-0-13-373762-2.
^"Characterizations of a linear subspace associated with Fourier series". MathOverflow. 2010-11-19. Retrieved 2014-08-08.
^Vanishing of Half the Fourier Coefficients in Staggered Arrays
^Siebert, William McC. (1985). Circuits, signals, and systems. MIT Press. p. 402. ISBN 978-0-262-19229-3.
^Marton, L.; Marton, Claire (1990). Advances in Electronics and Electron Physics. Academic Press. p. 369. ISBN 978-0-12-014650-5.
^Kuzmany, Hans (1998). Solid-state spectroscopy. Springer. p. 14. ISBN 978-3-540-63913-8.
^Pribram, Karl H.; Yasue, Kunio; Jibu, Mari (1991). Brain and perception. Lawrence Erlbaum Associates. p. 26. ISBN 978-0-89859-995-4.
William E. Boyce; Richard C. DiPrima (2005). Elementary Differential Equations and Boundary Value Problems (8th ed.). New Jersey: John Wiley & Sons, Inc. ISBN 0-471-43338-1.
Joseph Fourier, translated by Alexander Freeman (2003). The Analytical Theory of Heat. Dover Publications. ISBN 0-486-49531-0. 2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's work Théorie Analytique de la Chaleur, originally published in 1822.
Enrique A. Gonzalez-Velasco (1992). "Connections in Mathematical Analysis: The Case of Fourier Series". American Mathematical Monthly. 99 (5): 427–441. doi:10.2307/2325087. JSTOR 2325087.
Fetter, Alexander L.; Walecka, John Dirk (2003). Theoretical Mechanics of Particles and Continua. Courier. ISBN 978-0-486-43261-8.
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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.
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