↑ arvau

Wikipedia / Fourier series

A Fourier series () is a series expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. 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 by Joseph Fourier to find solutions to the heat 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 always converge. Well-behaved functions, for example smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric functions, described in .

The study of the convergence of Fourier series focus on the behaviors of the partial 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 a square wave.

Fourier series are closely related to the Fourier transform, a more general tool that can even find the frequency information for functions that are not periodic. Periodic functions can be identified with functions on a circle; for this reason Fourier series are the subject of Fourier analysis on the circle group. The Fourier transform is also part of Fourier 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 in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions in the series expansion. Many other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis.

HistoryHistorical development

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.

Independently of Fourier, astronomer Friedrich Wilhelm Bessel introduced Fourier series to solve Kepler's equation. His work was published in 1819, unaware of Fourier's work which remained unpublished until 1822.

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 sinusoidss. The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, shell theory, etc.

Beginnings

Joseph Fourier wrote

This immediately gives any coefficient ak of the trigonometric series for φ(y for term, which is 1.

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".

Fourier's motivation

, 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, then one can show that the stationary heat distribution (or the heat distribution after a long time has elapsed) is given by

is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of the equation from Analysis § Example cannot be written as a closed-form expression. This method of solving the heat problem was made possible by Fourier's work.

Other applications

Another application 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.

Definition

The Fourier series of a complex-valued on the real line, is defined as a trigonometric series such that the Fourier coefficients are complex numbers defined by the integral The series does not necessarily converge (in the pointwise For functions satisfying the Dirichlet sufficiency conditions, pointwise convergence holds. However, these are not necessary conditions and there are many theorems about different types of convergence of Fourier series (e.g. uniform convergence or mean convergence). The definition naturally extends to the Fourier series of a (periodic) distribution (also called Fourier-Schwartz series). in the distribution sense.

The process of determining the Fourier coefficients of a given function or signal is called analysis, while forming the associated trigonometric series (or its various approximations) is called synthesis.

Synthesis

partial sums:

. Therefore, the sinusoids have:

These series can represent functions that are just a sum of one or more frequencies in the harmonic spectrum, a trigonometric series can also represent the intermediate frequencies or non-sinusoidal functions because of the infinite number of terms.

Analysis

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 a discrete-time Fourier transform represents frequency instead of time. In general, the coefficients are determined by analysis whose domain of definition.

scale factor follows from substituting into and utilizing the orthogonality of the trigonometric system. The equivalence of and follows from Euler's formula resulting in: being the mean value. Conversely:

Example

. This is a particular instance of the Dirichlet theorem for Fourier series.

This example leads to a solution of the Basel problem.

Amplitude-phase form

is real-valued then the Fourier series can also be represented as is the amplitude is the phase shift harmonic.

The equivalence of and follows from the trigonometric identity which implies are the rectangular coordinates of a vector written in polar coordinates

and a cosine template,. The blue graph is the cross-correlation function, also known as a matched filter:

Common notations

and functional notation often replaces subscripting:

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:

has units of hertz. The "teeth" of the comb are spaced at multiples (i.e. harmonics, which is called the fundamental frequency can be recovered from this representation by an inverse Fourier transform:

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.

Table of common Fourier series

Some common pairs of periodic functions and their Fourier series coefficients are shown in the table below.

Table of basic transformation rules

This table shows some mathematical operations in the time domain and the corresponding effect in the Fourier series coefficients. Notation:

Properties

Symmetry relations

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:

Riemann–Lebesgue lemma

is integrable

Parseval's theorem

Plancherel's theorem

.

Convolution theorems

Derivative property

belongs to the function space.

  • .

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 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 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

is a compact Riemannian manifold, it has a Laplace–Beltrami operator. The Laplace–Beltrami operator is the differential operator that corresponds to Laplace operator 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 groupss. However, there is a straightforward generalization to Locally Compact Abelian (LCA) groups.

is noncompact, one obtains instead a Fourier integral. This generalization yields the usual Fourier transform.

Extensions

Fourier-Stieltjes series

is the space finite Borel measures is also referred to as a Fourier-Stieltjes transform.

This follows from an earlier and more concrete representation of a Radon measure (i.e. a locally finite Borel measure, given by F. Riesz is a function of bounded variation then the Fourier coefficients can be expressed by the Riemann-Stieltjes integral called the Fourier-Stieltjes coefficients. is a Radon measure, it is subject to the Lebesgue decomposition. the expression reduces to the original definition of the Fourier coefficients, hence a Fourier series is a Fourier-Stieltjes series.

forms the basis of the trigonometric moment problem.

The Fourier series can be generalized still further from measures to distributions then the series is sometimes described as a Fourier-Schwartz series.

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.

Fourier series on a 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, a discrete form of the Fourier 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.

Fourier series of a Bravais-lattice-periodic function

A three-dimensional Bravais lattice with the same periodicity as the Bravais lattice, i.e.. This situation frequently occurs in solid-state physics 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 as Bloch state.

in the previous section but in order to avoid double subscripts we note them as a function.

. This can be achieved by means of the reciprocal lattice the Kronecker delta appears as a Jacobian matrix

The unit cell being a parallelepiped

Hilbert space

As the trigonometric series is a special class of orthogonal system, Fourier series can naturally be defined in the context of Hilbert spacess. For example, the space of square-integrable functions. Its inner product This space is equipped with the orthonormal basis. Then the (generalized) Fourier series can be written as The sine-cosine form follows in a similar fashion. Indeed, the sines and cosines form an orthogonal set (where is the Kronecker delta. 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.

Fourier theorem proving convergence of Fourier series

In engineering. If a function is square-integrable, then the Fourier series converges to the function almost everywhere. It is possible to define Fourier coefficients for more general functions or distributions, in which case pointwise convergence often fails, and convergence in norm or weak convergence is usually studied.

The theorems proving that a Fourier series is a valid representation of any periodic function (that satisfies the Dirichlet conditions), and informal variations of them that do not specify the convergence conditions, are sometimes referred to generically as Fourier's theorem or the Fourier theorem.

Least squares property

The earlier :

is a trigonometric polynomial that can be generally expressed as:

Parseval's theorem implies that:

Convergence theorems

Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.

is continuous, and therefore bounded, it is square-integrable and its Fourier coefficients are square-summable. Then, by the Cauchy–Schwarz inequality,

is absolutely summable:

satisfies a Hölder condition. In the absolutely summable case, the inequality:

proves uniform convergence.

, to more sophisticated results such as Carleson's theorem function converges almost everywhere.

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.

It is possible to give explicit examples of a continuous function whose Fourier series diverges at 0: for instance, the even and 2π-periodic function f defined for all x in [0,π] by

Because the function is even the Fourier series contains only cosines:

The coefficients are:

As for some and this builds up to around

in the th 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.

See also

Notes

References

Bibliography