Fourier series

Fourier series

Overview
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Fourier series decomposes periodic functions
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

 or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

  (or complex exponentials). The study of Fourier series is a branch of Fourier analysis.

The Fourier series is named in honour of Joseph Fourier
Joseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...

 (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

, Jean le Rond d'Alembert
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. He was also co-editor with Denis Diderot of the Encyclopédie...

, and Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

.
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Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Fourier series decomposes periodic functions
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

 or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines
Sine wave
The sine wave or sinusoid is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure mathematics, as well as physics, signal processing, electrical engineering and many other fields...

  (or complex exponentials). The study of Fourier series is a branch of Fourier analysis.

The Fourier series is named in honour of Joseph Fourier
Joseph Fourier
Jean Baptiste Joseph Fourier was a French mathematician and physicist best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's Law are also named in his honour...

 (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler
Leonhard Euler
Leonhard Euler was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion...

, Jean le Rond d'Alembert
Jean le Rond d'Alembert
Jean-Baptiste le Rond d'Alembert was a French mathematician, mechanician, physicist, philosopher, and music theorist. He was also co-editor with Denis Diderot of the Encyclopédie...

, and Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

. Fourier introduced the series for the purpose of solving the heat equation
Heat equation
The heat equation is an important partial differential equation which describes the distribution of heat in a given region over time...

 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 in 1822.

The heat equation is a partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...

. 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
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

 or cosine wave. These simple solutions are now sometimes called eigensolutions
Eigenvalue, eigenvector and eigenspace
The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix...

. Fourier's idea was to model a complicated heat source as a superposition (or linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...

) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...

. 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
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 and integral
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...

 in the early nineteenth century. Later, Dirichlet and 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
Electrical engineering
Electrical engineering is a field of engineering that generally deals with the study and application of electricity, electronics and electromagnetism. The field first became an identifiable occupation in the late nineteenth century after commercialization of the electric telegraph and electrical...

, vibration
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

 analysis, acoustics
Acoustics
Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics...

, optics
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...

, signal processing
Signal processing
Signal processing is an area of systems engineering, electrical engineering and applied mathematics that deals with operations on or analysis of signals, in either discrete or continuous time...

, image processing
Image processing
In electrical engineering and computer science, image processing is any form of signal processing for which the input is an image, such as a photograph or video frame; the output of image processing may be either an image or, a set of characteristics or parameters related to the image...

, quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, econometrics
Econometrics
Econometrics has been defined as "the application of mathematics and statistical methods to economic data" and described as the branch of economics "that aims to give empirical content to economic relations." More precisely, it is "the quantitative analysis of actual economic phenomena based on...

, thin-walled shell theory, etc.

Revolutionary article


This immediately gives any coefficient of the trigonometrical series for for any function which has such an expansion. It works because if has such an expansion, then (under suitable convergence assumptions) the integral
can be carried out term-by-term. But all terms involving for vanish when integrated from −1 to 1, leaving only the kth term.

In these few lines, which are close to the modern formalism
Formalism (mathematics)
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be thought of as statements about the consequences of certain string manipulation rules....

 used in Fourier series, Fourier revolutionized both mathematics and physics. Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...

 and Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...

, Fourier believed that such trigonometric series could represent arbitrary functions. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

s, and harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

.

When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange
Joseph Louis Lagrange
Joseph-Louis Lagrange , born Giuseppe Lodovico Lagrangia, was a mathematician and astronomer, who was born in Turin, Piedmont, lived part of his life in Prussia and part in France, making significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics...

, Laplace, Malus
Étienne-Louis Malus
- External links :...

 and Legendre
Adrien-Marie Legendre
Adrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...

, 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
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...

 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
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

. A Fourier series, however, can be used only for periodic functions, or for functions on a bounded (compact) interval.

Definition


In this section, ƒ(x) denotes a function of the real variable x. This function is usually taken to be periodic
Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π radians. Periodic functions are used throughout science to describe oscillations,...

, of period 2π, which is to say that ƒ(x + 2π) = ƒ(x), for all real numbers x. We will attempt to write such a function as an infinite sum, or series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....

 of simpler 2π–periodic functions. We will start by using an infinite sum of sine
Sine
In mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....

 and cosine functions on the interval [−π, π], as Fourier did (see the quote above), and we will then discuss different formulations and generalizations.

Fourier's formula for 2π-periodic functions using sines and cosines


For a periodic function ƒ(x) that is integrable on [−π, π], the numbers


and


are called the Fourier coefficients of ƒ. One introduces the partial sums of the Fourier series for ƒ, often denoted by


The partial sums for ƒ are trigonometric polynomial
Trigonometric polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin and cos with n a natural number. The coefficients may be taken as real numbers, for real-valued functions...

s. One expects that the functions SN ƒ approximate the function ƒ, and that the approximation improves as N tends to infinity. The infinite sum


is called the Fourier series of ƒ. These trigonometric functions can themselves be expanded, using multiple angle formulae.

The Fourier series does not always converge, and even when it does converge for a specific value x0 of x, the sum of the series at x0 may differ from the value ƒ(x0) of the function. It is one of the main questions in harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...

 to decide when Fourier series converge, and when the sum is equal to the original function. If a function is square-integrable
Square-integrable function
In mathematics, a quadratically integrable function, also called a square-integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite...

 on the interval [−π, π], then the Fourier series converges to the function at almost every point. In engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

 applications, the Fourier series is generally presumed to converge everywhere except at discontinuities, since the functions encountered in engineering are more well behaved than the ones that mathematicians can provide as counter-examples to this presumption. In particular, the Fourier series converges absolutely and uniformly to ƒ(x) whenever the derivative of ƒ(x) (which may not exist everywhere) is square integrable. See Convergence of Fourier series
Convergence of Fourier series
In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics...

.

It is possible to define Fourier coefficients for more general functions or distributions, in such cases convergence in norm or weak convergence
Weak convergence (Hilbert space)
In mathematics, weak convergence in a Hilbert space is convergence of a sequence of points in the weak topology.-Definition:A sequence of points in a Hilbert space H is said to converge weakly to a point x in H if...

 is usually of interest.

Example 1: a simple 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 the Fourier series converges to ƒ(x) at every point x where ƒ is differentiable, and therefore:
When x = π, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of ƒ at x = π. This is a particular instance of the Dirichlet theorem for Fourier series.

Example 2: Fourier's motivation


One notices that the Fourier series expansion of our function in example 1 looks much less simple than the formula ƒ(x) = x, and so it is not immediately apparent why one would need this Fourier series. While there are many applications, we cite Fourier's motivation of solving the heat equation. For example, consider a metal plate in the shape of a square whose side measures π meters, with coordinates (x, y) ∈ [0, π] × [0, π]. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by y = π, is maintained at the temperature gradient T(x, π) = x degrees Celsius, for x in (0, π), then one can show that the stationary heat distribution (or the heat distribution after a long period of time has elapsed) is given by


Here, sinh is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of   by sinh(ny)/sinh(nπ). While our example function f(x) seems to have a needlessly complicated Fourier series, the heat distribution T(x, y) is nontrivial. The function T cannot be written as a closed-form expression
Closed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions...

. This method of solving the heat problem was made possible by Fourier's work.

Other applications


Another application of this Fourier series is to solve the Basel problem
Basel problem
The Basel problem is a famous problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735. Since the problem had withstood the attacks of the leading mathematicians of the day, Euler's solution brought him immediate...

 by using Parseval's theorem
Parseval's theorem
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later...

. The example generalizes and one may compute ζ(2n), for any positive integer n.

Exponential Fourier series


We can use Euler's formula
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function...

,


where i is the imaginary unit
Imaginary unit
In mathematics, the imaginary unit allows the real number system ℝ to be extended to the complex number system ℂ, which in turn provides at least one root for every polynomial . The imaginary unit is denoted by , , or the Greek...

, to give a more concise formula:


The Fourier coefficients are then given by:


The Fourier coefficients an, bn, cn are related via

and


The notation cn is inadequate for discussing the Fourier coefficients of several different functions. Therefore it is customarily replaced by a modified form of ƒ (in this case), such as F or   and functional notation often replaces subscripting.  Thus:


In engineering, particularly when the variable x represents time, the coefficient sequence is called a frequency domain
Frequency domain
In electronics, control systems engineering, and statistics, frequency domain is a term used to describe the domain for analysis of mathematical functions or signals with respect to frequency, rather than time....

 representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.

Fourier series on a general interval [a, a + τ]


The following formula, with appropriate complex-valued coefficients G[n], is a periodic function with period τ on all of R:


If a function is square-integrable in the interval [a, a + τ], it can be represented in that interval by the formula above.  I.e., when the coefficients are derived from a function, h(x), as follows:


then g(x) will equal h(x) in the interval [a,a+τ ]. It follows that if h(x) is τ-periodic, then:
  • g(x) and h(x) are equal everywhere, except possibly at discontinuities, and
  • a is an arbitrary choice. Two popular choices are a = 0, and a = −τ/2.


Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb
Dirac comb
In mathematics, a Dirac comb is a periodic Schwartz distribution constructed from Dirac delta functions...

:


where variable ƒ represents a continuous frequency domain. When variable x has units of seconds, ƒ has units of hertz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of 1/τ, which is called the fundamental frequency
Fundamental frequency
The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the lowest frequency of a periodic waveform. In terms of a superposition of sinusoids The fundamental frequency, often referred to simply as the fundamental and abbreviated f0, is defined as the...

.  g(x) can be recovered from this representation by an inverse Fourier transform
Fourier inversion theorem
In mathematics, Fourier inversion recovers a function from its Fourier transform. Several different Fourier inversion theorems exist.Sometimes the following expression is used as the definition of the Fourier transform:...

:


The function G(ƒ) is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.

Fourier series on a square


We can also define the Fourier series for functions of two variables x and y in the square [−π, π]×[−π, π]:


Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series on the square is in image compression
Image compression
The objective of image compression is to reduce irrelevance and redundancy of the image data in order to be able to store or transmit data in an efficient form.- Lossy and lossless compression :...

. In particular, the jpeg
JPEG
In computing, JPEG . The degree of compression can be adjusted, allowing a selectable tradeoff between storage size and image quality. JPEG typically achieves 10:1 compression with little perceptible loss in image quality....

 image compression standard uses the two-dimensional discrete cosine transform
Discrete cosine transform
A discrete cosine transform expresses a sequence of finitely many data points in terms of a sum of cosine functions oscillating at different frequencies. DCTs are important to numerous applications in science and engineering, from lossy compression of audio and images A discrete cosine transform...

, which is a Fourier transform using the cosine basis functions.

Hilbert space interpretation


In the language of Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

s, the set of functions is an orthonormal basis
Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...

 for the space of square-integrable functions of . This space is actually a Hilbert space with an inner product given for any two elements f and g by:


The basic Fourier series result for Hilbert spaces can be written as


This corresponds exactly to the complex exponential formulation given above. The version with sines and cosines is also justified with the Hilbert space interpretation. Indeed, the sines and cosines form an orthogonal set:


(where is the Kronecker delta), and


furthermore, the sines and cosines are orthogonal to the constant function 1. An orthonormal basis for L2([−π, π]) consisting of real functions is formed by the functions 1, and √2 cos(n  x),  √2 sin(n x) for n = 1, 2,...  The density of their span is a consequence of the Stone–Weierstrass theorem, but follows also from the properties of classical kernels like the Fejér kernel.

Properties


We say that ƒ belongs to    if ƒ is a 2π-periodic function on R which is k times differentiable, and its kth derivative is continuous.
  • If ƒ is a 2π-periodic odd function, then   for all n.

  • If ƒ is a 2π-periodic even function, then   for all n.

  • If ƒ is integrable, , and This result is known as the Riemann–Lebesgue lemma.


  • 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 tends to zero, we have that tends to zero, which means that the Fourier coefficients converge to zero faster than the kth power of n.

  • Parseval's theorem
    Parseval's theorem
    In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later...

    . If , then .

  • Plancherel's theorem. If are coefficients and then there is a unique function such that for every n.

  • The first convolution theorem
    Convolution theorem
    In mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain...

     states that if ƒ and g are in L1([−π, π]), then , where ƒ ∗ g denotes the 2π-periodic convolution
    Convolution
    In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

     of ƒ and g. (The factor is not necessary for 1-periodic functions.)

  • The second convolution theorem
    Convolution theorem
    In mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a convolution is the pointwise product of Fourier transforms. In other words, convolution in one domain equals point-wise multiplication in the other domain...

     states that .

  • The Poisson summation formula
    Poisson summation formula
    In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples...

     states that the periodic summation of a function,    has a Fourier series representation whose coefficients are proportional to discrete samples of the continuous Fourier transform
    Fourier transform
    In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

     of :.
 
Similarly, the periodic summation of has a Fourier series representation whose coefficients are proportional to discrete samples of , a fact which provides a pictorial understanding of aliasing and the famous sampling theorem
Nyquist–Shannon sampling theorem
The Nyquist–Shannon sampling theorem, after Harry Nyquist and Claude Shannon, is a fundamental result in the field of information theory, in particular telecommunications and signal processing. Sampling is the process of converting a signal into a numeric sequence...

.
  • Also see Relations between Fourier transforms and Fourier series
    Relations between Fourier transforms and Fourier series
    In the mathematical field of harmonic analysis, the continuous Fourier transform has very precise relations with Fourier series. It is also closely related to the discrete-time Fourier transform and the discrete Fourier transform ....

    .

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
Compact group
In mathematics, a compact group is a topological group whose topology is compact. Compact groups are a natural generalisation of finite groups with the discrete topology and have properties that carry over in significant fashion...

. Typical examples include those classical group
Classical group
In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type...

s 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 convolution
Convolution
In mathematics and, in particular, functional analysis, convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions. Convolution is similar to cross-correlation...

s 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
Peter–Weyl theorem
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G...

, 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
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 Riemannian manifold
Riemannian manifold
In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

, it has a Laplace–Beltrami operator. The Laplace–Beltrami operator is the differential operator that corresponds to Laplace operator
Laplace operator
In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...

 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 L2(X), where X is a Riemannian manifold. The Fourier series converges in ways similar to the [−π, π] case. A typical example is to take X to be the sphere with the usual metric, in which case the Fourier basis consists of spherical harmonics
Spherical harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...

.

Locally compact Abelian groups


The generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups. However, there is a straightfoward generalization to Locally Compact Abelian (LCA) groups.

This generalizes the Fourier transform to L1(G) or L2(G), where G is an LCA group. If G is compact, one also obtains a Fourier series, which converges similarly to the [−π, π] case, but if G is noncompact, one obtains instead a Fourier integral. This generalization yields the usual Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...

 when the underlying locally compact Abelian group is .

Approximation and convergence of Fourier series


An important question for the theory as well as applications is that of convergence. In particular, it is often necessary in applications to replace the infinite series   by a finite one,


This is called a partial sum. We would like to know, in which sense does (SN ƒ)(x) converge to ƒ(x) as N tends to infinity.

Least squares property


We say that p is a trigonometric polynomial
Trigonometric polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin and cos with n a natural number. The coefficients may be taken as real numbers, for real-valued functions...

 of degree N when it is of the form


Note that SN ƒ is a trigonometric polynomial of degree N. Parseval's theorem implies that

Theorem. The trigonometric polynomial SN ƒ is the unique best trigonometric polynomial of degree N approximating ƒ(x), in the sense that, for any trigonometric polynomial of degree N, we have 

Here, the Hilbert space norm is

Convergence


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

Theorem. If ƒ belongs to L2([−π, π]), then the Fourier series converges to ƒ in L2([−π, π]), that is,  converges to 0 as N goes to infinity.

We have already mentioned that if ƒ is continuously differentiable, then    is the nth Fourier coefficient of the derivative ƒ′. It follows, essentially from the Cauchy–Schwarz inequality
Cauchy–Schwarz inequality
In mathematics, the Cauchy–Schwarz inequality , is a useful inequality encountered in many different settings, such as linear algebra, analysis, probability theory, and other areas...

, that the Fourier series of ƒ is absolutely summable. The sum of this series is a continuous function, equal to ƒ, since the Fourier series converges in the mean to ƒ:

Theorem. If  , then the Fourier series converges to ƒ uniformly (and hence also pointwise
Pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.-Definition:...

.)

This result can be proven easily if ƒ is further assumed to be C2, since in that case tends to zero as . More generally, the Fourier series is absolutely summable, thus converges uniformly to ƒ, provided that ƒ satisfies a Hölder condition
Hölder condition
In mathematics, a real or complex-valued function ƒ on d-dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are nonnegative real constants C, \alpha , such that...

 of order α > ½. In the absolutely summable case, the inequality    proves uniform convergence.

Many other results concerning the convergence of Fourier series
Convergence of Fourier series
In mathematics, the question of whether the Fourier series of a periodic function converges to the given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics...

 are known, ranging from the moderately simple result that the series converges at x if ƒ is differentiable at x, to Lennart Carleson
Lennart Carleson
Lennart Axel Edvard Carleson is a Swedish mathematician, known as a leader in the field of harmonic analysis.-Life:He was a student of Arne Beurling and received his Ph.D. from Uppsala University in 1950...

's much more sophisticated result that the Fourier series of an L2 function actually converges almost everywhere
Almost everywhere
In measure theory , a property holds almost everywhere if the set of elements for which the property does not hold is a null set, that is, a set of measure zero . In cases where the measure is not complete, it is sufficient that the set is contained within a set of measure zero...

.

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".

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
Uniform boundedness principle
In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornerstones of the field...

 yields a simple non-constructive proof of this fact.

In 1922, Andrey Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...

 published an article entitled "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

  • Gibbs phenomenon
    Gibbs phenomenon
    In mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity: the nth partial sum of the Fourier series has large...

  • Laurent series
    Laurent series
    In mathematics, the Laurent series of a complex function f is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where...

     — the substitution q = eix transforms a Fourier series into a Laurent series, or conversely. This is used in the q-series expansion of the j-invariant
    J-invariant
    In mathematics, Klein's j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half-plane of complex numbers.We haveThe modular discriminant \Delta is defined as \Delta=g_2^3-27g_3^2...

    .
  • Sturm–Liouville theory
  • ATS theorem
    ATS theorem
    In mathematics, the ATS theorem is the theorem on the approximation of atrigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful....

  • Discrete Fourier transform
    Discrete Fourier transform
    In mathematics, the discrete Fourier transform is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, of the original function...

  • Spectral theory
    Spectral theory
    In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of...

  • Fejér's theorem
    Fejér's theorem
    In mathematics, Fejér's theorem, named for Hungarian mathematician Lipót Fejér, states that if f:R → C is a continuous function with period 2π, then the sequence of Cesàro means of the sequence of partial sums of the Fourier series of f converges uniformly to f on...

  • Dirichlet kernel

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
    Felix Klein
    Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...

    , 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