Fourier series

Fourier transforms
Continuous Fourier transform
Fourier series
Discrete Fourier transform
Discrete-time Fourier transform
Related transforms

In mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating functions, namely sines and cosines. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and In Mathematics, the discrete Fourier transform (DFT is one of the specific forms of Fourier analysis. In Mathematics, the discrete-time Fourier transform (DTFT is one of the specific forms of Fourier analysis. This is a list of Linear transformations of functions related to Fourier analysis. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and The study of Fourier series is a branch of Fourier analysis. In mathematics Fourier analysis is a subject area which grew out of the study of Fourier series Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate, it led to a revolution in mathematics, forcing mathematicians to reexamine the foundations of mathematics and leading to many modern theories such as Lebesgue integration. Jean Baptiste Joseph Fourier ( March 21, 1768 &ndash May 16, 1830) was a French Mathematician and Physicist The heat equation is an important Partial differential equation which describes the distribution of Heat (or variation in temperature in a given region over time In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of

The heat equation is a partial differential equation. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i Prior to Fourier's work, there was no known solution to the heat equation in a general situation. 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. In Mathematics, linear combinations are a concept central to Linear algebra and related fields of mathematics This superposition or linear combination is called the Fourier series.

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. The heat equation is an important Partial differential equation which describes the distribution of Heat (or variation in temperature in a given region over time The basic results are very easy to understand using the modern theory.

The Fourier series has many applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, etc. Electrical engineering, sometimes referred to as electrical and electronic engineering, is a field of Engineering that deals with the study and application of Oscillation is the repetitive variation typically in Time, of some measure about a central value (often a point of Equilibrium) or between two or more different states Acoustics is the interdisciplinary science that deals with the study of Sound, Ultrasound and Infrasound (all mechanical waves in gases liquids and solids Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as Image processing is any form of Signal processing for which the input is an image such as photographs or frames of video the output of image processing can be either an image

Historical development

Fourier series are named in honor of Joseph Fourier (1768-1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Madhava, Nilakantha Somayaji, Jyesthadeva, Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. Jean Baptiste Joseph Fourier ( March 21, 1768 &ndash May 16, 1830) was a French Mathematician and Physicist Mādhava of Sangamagrama (born as Irinjaatappilly Madhavan Namboodiri) (c Nilakantha Somayaji ( Malayalam: നീലകണ്ഠ സോമയാജി hindi नीलकण्ठ सोमयाजि (1444-1544 from Kerala, was a major Jyestadeva (ജ്യേഷ്ഠദേവ(ന് (1500 &ndash 1575 was an astronomer of the Kerala school founded by Madhava of Sangamagrama and Daniel Bernoulli ( Groningen, 29 January 1700 &ndash 27 July 1782 was a Dutch - Swiss Mathematician, who is particularly remembered for his applications He applied this technique to find the solution of the heat equation, publishing his initial results in 1807 and 1811, and publishing his Théorie analytique de la chaleur in 1822. The heat equation is an important Partial differential equation which describes the distribution of Heat (or variation in temperature in a given region over time

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 (for example, one wonderedif a function defined on two intervals with two different formulas was still a function). The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Later, Dirichlet and Riemann expressed Fourier's results with greater precision and formality. Johann Peter Gustav Lejeune Dirichlet (ləʒœn diʀiçle February 13, 1805 &ndash May 5, 1859) was a German Mathematician

A revolutionary article

“

$\varphi(y)=a\cos\frac{\pi y}{2}+a'\cos 3\frac{\pi y}{2}+a''\cos5\frac{\pi y}{2}+\cdots.$

Multiplying both sides by $\cos(2i+1)\frac{\pi y}{2}$ , and then integrating from $y = - 1$ to $y = + 1$ yields:

$a_i=\int_{-1}^1\varphi(y)\cos(2i+1)\frac{\pi y}{2}\,dy.$

”

—Joseph Fourier, Mémoire sur la propagation de la chaleur dans les corps solides, pp. 218--219. ^[1]

In these few lines, which are surprisingly close to the modern formalism used in Fourier series, Fourier unwittingly 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 arbitrary functions. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German While this is not true, the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis. In the absence of a more specific context convergence denotes the approach toward a definite value as time goes on or to a definite point a common view or opinion or In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y. 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

When Fourier submitted his paper in 1807, the committee (composed of no lesser mathematicians than Lagrange, Laplace, Malus and Legendre, among others) concluded: . Etienne-Louis Malus (23 July 1775 &ndash 24 February 1812 was a French officer, Engineer, Physicist, and Mathematician. Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. . . 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.

The birth of harmonic analysis

Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available at the time Fourier completed his original work. Fourier originally defined the Fourier series for real-valued functions of real arguments, and using the sine and cosine functions as the basis set for the decomposition. Basis vector redirects here For basis vector in the context of crystals see Crystal structure.

Many other Fourier-related transforms have since been defined, extending the initial idea to other applications. This is a list of Linear transformations of functions related to Fourier analysis. This general area of inquiry is now sometimes called 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

Definition

In this section, f(x) denotes a function of the real variable x. This function is usually taken to be periodic, of period 2π, which is to say that f(x+2π) = f(x), for all real numbers x. In Mathematics, a periodic function is a function that repeats its values after some definite period has been added to its Independent variable We will show how to write such a function as an infinite sum, or series. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with We will start by using an infinite sum of sine and cosine functions of 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 2π-periodic function f(x) the numbers

$a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos(nx)\, dx$

and

$b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \sin(nx)\, dx$

are called the Fourier coefficients of f. The infinite sum

$f(x) = \frac{a_0}{2} +\sum_{n=1}^{\infty}[a_n \cos(nx) + b_n \sin(nx)]$

is the Fourier series for f on the interval [-π,π]. In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with The Fourier series does not always converge, so there may not be equality in the formula above. It is one of the main questions in Harmonic analysis to decide when equality holds. 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 If a function is square-integrable on the interval [-π,π], then it can be represented in that interval by the previous formula. In Mathematics, an integrable function is a function whose Integral exists

Example: a simple Fourier series

Plot of a periodic identity function - a sawtooth wave. The sawtooth wave (or saw wave) is a kind of Non-sinusoidal waveform.

Animated plot of the first five successive partial Fourier series.

We now use the formulae above to give a Fourier series expansion of a very simple function. Consider a sawtooth function (as depicted in the figure):

$f(x) = x, \quad \mathrm{for} \quad -\pi < x < \pi,$

$f(x + 2\pi) = f(x), \quad \mathrm{for} \quad -\infty < x < \infty.$

In this case, the Fourier coefficients are given by

$\begin{align}a_n &{} = \frac{1}{\pi}\int_{-\pi}^{\pi}x \cos(nx)\,dx = 0. \\b_n &{}= \frac{1}{\pi}\int_{-\pi}^{\pi} x \sin(nx)\, dx = 2\frac{(-1)^{n+1}}{n}.\end{align}$

And therefore:

$\begin{align}f(x) &= \frac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n\cos\left(nx\right)+b_n\sin\left(nx\right)\right] \\&=2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} \sin(nx), \quad \mathrm{for} \quad -\infty < x < \infty .\end{align}$		(Eq. 1)

Heat distribution in a metal plate, using Fourier's method.

One notices that the Fourier series expansion of our function looks much less simple than the formula f(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 $(x,y) \in [0,\pi] \times [0,\pi]$ . 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

$T(x,y) = 2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx) {\sinh(ny) \over \sinh(n\pi)}.$

Here, sinh is the hyperbolic sine function. The Celsius Temperature scale was previously known as the centigrade scale. In Mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular functions This solution of the heat equation is obtained by multiplying each term of (Eq. 1) 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. This method of solving the heat problem was only made possible by Fourier's work.

Another application of this Fourier series is to solve the Basel problem by using Parseval's theorem. The Basel problem is a famous problem in Number theory, first posed by Pietro Mengoli in 1644 and solved by Leonhard Euler in 1735 In Mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely that the sum (or integral of the square The example generalizes and one may compute zeta(2n), for any positive integer n. In Mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of great significance in

The modern version using complex exponentials

We can use Euler's formula, $e i n x = cos(n x) + i sin(n x)$ , where $i$ is the imaginary unit, to give a more concise formula:

$f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inx}.$

The Fourier coefficients are then given by:

$c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) e^{-inx}\, dx.$

The Fourier coefficients $a n, b n, c n$ are related via

a n = c n + c - n

for $n=0,1,2,\dots,$

and

b n = i (c n - c - n)

for $n=1,2,\dots$

The notation $c n$ is inadequate for discussing the Fourier coefficients of several different functions. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic Definition By definition the imaginary unit i is one solution (of two of the Quadratic equation Therefore it is customarily replaced by a modified form of $f\,$ (in this case), such as $F\,$ or $\hat{f},$ and functional notation often replaces subscripting. Thus:

$\begin{align}f(x) &= \sum_{n=-\infty}^{\infty} \hat{f}(n)\cdot e^{inx} \\&= \sum_{n=-\infty}^{\infty} F[n]\cdot e^{inx} \quad \mbox{(engineering)}.\end{align}$

In various fields of science, the sequence has other names, such as characteristic function (probability theory). In Probability theory, the characteristic function of any Random variable completely defines its Probability distribution. In engineering, particularly when variable x represents time, the sequence is called a frequency domain representation. Frequency domain is a term used to describe the analysis of Mathematical functions or signals with respect to frequency 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,b]

Let G[0], G[±1], G[±2], … be real or complex coefficients. The Fourier series:

$g(x)=\sum_{n=-\infty}^\infty G[n]\cdot e^{i 2\pi \frac{n}{\tau} x}\,$

is a periodic function, whose period is $\tau\,$ on the domain $\mathbb{R}.$ If a function is square-integrable in the interval $[a,\ a+\tau],$ it can be represented in that interval by the formula above. In Mathematics, an integrable function is a function whose Integral exists If g(x) is integrable, then the Fourier coefficients are given by:

$G[n] = \frac{1}{\tau}\int_a^{a+\tau} g(x)\cdot e^{-i 2\pi \frac{n}{\tau} x}\, dx.$

Note that if the function to be represented is also $\tau\,$ -periodic, then $a\,$ is an arbitrary choice. Two popular choices are $a=0\,$ and $a=-\tau/2.\,$

Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:

$G(f) \ \stackrel{\mathrm{def}}{=} \ \sum_{n=-\infty}^{\infty} G[n]\cdot \delta \left(f-\frac{n}{\tau}\right)$

where variable $f\,$ represents a continuous frequency domain. In Mathematics, a Dirac comb (also known as an impulse train and sampling function in Electrical engineering) is a periodic When variable $x\,$ has units of seconds, $f\,$ has units of hertz. The hertz (symbol Hz) is a measure of Frequency, informally defined as the number of events occurring per Second. The "teeth" of the comb are spaced at multiples (i. e. harmonics) of $1/\tau,\,$ which is called the fundamental frequency. In Acoustics and Telecommunication, the harmonic of a Wave is a component Frequency of the signal that is an Integer The fundamental tone, often referred to simply as the fundamental and abbreviated fo, is the lowest frequency in a harmonic series. The original $g(x)\,$ can be recovered from this representation by an inverse Fourier transform. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and ^[2] The function $G(f)\,$ is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent. ^[3]

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]:

$f(x,y) = \sum_{j,k \in \mathbb{Z}} c_{j,k}e^{ijx}e^{iky},$

$c_{j,k} = {1 \over 4 \pi^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} f(x,y) e^{-ijx}e^{-iky}\, dx \, dy\ .$

Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series of the square is in image compression. Image compression is the application of Data compression on Digital images In effect the objective is to reduce redundancy of the image data in order to be able to In particular, the jpeg image compression standard uses the two-dimensional discrete cosine transform, which is a Fourier transform using the cosine basis functions. A discrete cosine transform ( DCT) expresses a sequence of finitely many data points in terms of a sum of Cosine functions oscillating at different frequencies

Hilbert space interpretation

Main article: Hilbert space

In the language of Hilbert spaces, the set of functions $\{ e_n = e^{i n x},n\in\mathbb{Z}\}$ is an orthonormal basis for the space $L 2 ([ - π,π])$ of square-integrable functions of $[ - π,π]$ . This article assumes some familiarity with Analytic geometry and the concept of a limit. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, an orthonormal basis of an Inner product space V (i This space is actually a Hilbert space with an inner product given by:

$\langle f, g \rangle \ \stackrel{\mathrm{def}}{=} \ \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\overline{g(x)}\,dx.$

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

$f=\sum_{n=-\infty}^{\infty} \langle f,e_n \rangle e_n.$

This corresponds exactly to the complex exponential formulation given above. This article assumes some familiarity with Analytic geometry and the concept of a limit. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. The version with sines and cosines is also justified with the Hilbert space interpretation. Clearly, the sines and cosines form an orthonormal set:

$\int_{-\pi}^{\pi} \cos(mx)\, \cos(nx)\, dx = \pi \delta_{mn},$

$\int_{-\pi}^{\pi} \sin(mx)\, \sin(nx)\, dx = \pi \delta_{mn}$

(where $δ m n$ is the Kronecker delta), and

$\int_{-\pi}^{\pi} \cos(mx)\, \sin(nx)\, dx = 0.$

The density of their span is a consequence of the Stone-Weierstrass theorem. In Linear algebra, two vectors in an Inner product space are orthonormal if they are orthogonal and both of unit length In Mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker ( 1823 - 1891) is a function of two

Properties

We say that $f \in C^k(\mathbb{T})$ if $f$ is a function of $\mathbb{R}$ which is $k$ times differentiable, its kth derivative is continuous, and is $2π$ -periodic.

If f is a $2π$ -periodic odd function, then $a n = 0$ for all $n$ . In Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive

If f is a $2π$ -periodic even function, then $b n = 0$ for all $n$ . In Mathematics, even functions and odd functions are functions which satisfy particular Symmetry relations with respect to taking Additive

If f is integrable, $\lim_{|n|\rightarrow \infty}\hat{f}(n)=0$ , $\lim_{n\rightarrow +\infty}a_n=0$ and $\lim_{n\rightarrow +\infty}b_n=0.$ This result is known as the Riemann-Lebesgue Lemma. In Mathematics, the Riemann-Lebesgue lemma (one of its special cases is also called Mercer's theorem) is of importance in Harmonic analysis and Asymptotic

If $f \in C^1(\mathbb{T})$ , then the Fourier coefficients $\hat{f'}(n)$ of the derivative $f'(t)$ can be expressed in terms of the Fourier coefficients $\hat{f}(n)$ of the function $f (t)$ , via the formula $\hat{f'}(n) = in \hat{f}(n)$ .

If $f \in C^k(\mathbb{T})$ , then $\widehat{f^{(k)}}(n) = (in)^k \hat{f}(n)$ . In particular, since $\widehat{f^{(k)}}(n)$ tends to zero, we have that $|n|^k\hat{f}(n)$ tends to zero, which means that the Fourier coefficients converge to zero faster than the kth power of n.

THEOREM (Parseval). In Mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely that the sum (or integral of the square If $f \in L^2([-\pi,\pi])$ , then $\sum_{n=-\infty}^{\infty} |\hat{f}(n)|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi} |f(x)|^2 \, dx$ .

THEOREM (Plancherel). In Mathematics, the Plancherel theorem is a result in Harmonic analysis, first proved by Michel Plancherel. If $c_0,\, c_{\pm 1},\, c_{\pm 2},\ldots$ are coefficients and $\sum_{n=-\infty}^\infty |c_n|^2 < \infty$ then there is a unique function $f\in L^2([-\pi,\pi])$ such that $\hat{f}(n) = c_n$ for every $n$ .

The Convolution theorem states that if f and g are in $L 2 ([ - π,π])$ , then $\widehat{f*g}(n) = \hat{f}(n)\hat{g}(n)$ , where $f * g$ denotes the $2π$ -periodic convolution of f and g. In Mathematics, the convolution theorem states that under suitableconditions the Fourier transform of a Convolution is the Pointwise product In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and

General case

There are many possible avenues for generalizing Fourier series. The study of Fourier series and its generalizations is called 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

Generalized functions

Main articles: Generalized function and Distribution (mathematics)

One can extend the notion of Fourier coefficients to functions which are not square-integrable, and even to objects which are not functions. In Mathematics, generalized functions are objects generalizing the notion of functions There is more than one recognised theory In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions This is very useful in engineering and applications because we often need to take the Fourier transform of a Dirac delta function. The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. The Dirac delta $δ$ is not actually a function, it is a measure but it still has a Fourier transform, and $\hat{\delta}(n)={1 \over 2\pi}$ for every $n$ . This generalization enlarges the domain of definition of the Fourier transform from $L 2 ([ - π,π])$ to a superset of $L 2$ . The Fourier series converges weakly. In Mathematics, weak convergence may refer to The weak Convergence of random variables of a Probability distribution.

Compact groups

Main articles: Compact group and Lie group

One of the interesting properties of the Fourier transform which we have mentioned, is that it carries convolutions to pointwise products. In Mathematics, a compact ( topological, often understood group is a Topological group whose Topology is Compact. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group If that is the property which we seek to preserve, one can produce Fourier series on any compact group. In Mathematics, a compact ( topological, often understood group is a Topological group whose Topology is Compact. Typical examples include those classical groups that are compact. The classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces There is a certain leeway in using the term This generalizes the Fourier transform to all spaces of the form $L 2 (G)$ , where $G$ is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products. In Mathematics and in particular Functional analysis, convolution is a mathematical operation on two functions f and The Fourier series exists and converges in similar ways to the $[ - π,π]$ case.

Riemannian manifolds

The atomic orbitals of chemistry are spherical harmonics and can be used to produce Fourier series on the sphere. An atomic orbital is a Mathematical function that describes the wave-like behavior of an electron in an atom Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

Main articles: Laplace operator and Riemannian manifold

If the domain is not a group, then there is no intrinsically defined convolution. In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M However, if $X$ is a compact Riemannian manifold, it has a Laplace-Beltrami operator. In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M Since the Laplace-Beltrami operator is the differential operator that corresponds to Laplace operator for the Riemannian manifold $X$ . In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after Then, by analogy, one can then consider heat equations on $X$ . Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace-Beltrami operator as a basis. This generalizes Fourier series to spaces of the type $L 2 (X)$ , where $X$ is a Riemannian manifold. The Fourier series converges in ways similar to the $[ - π,π]$ 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. In Mathematics, the spherical harmonics are the angular portion of an Orthogonal set of solutions to Laplace's equation represented in a system of

Locally compact Abelian groups

Main article: Pontryagin duality

The generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups. In Mathematics, in particular in Harmonic analysis and the theory of Topological groups Pontryagin duality explains the general properties of the Fourier However, there is a straightfoward generalization to Locally Compact Abelian (LCA) groups.

This generalizes the Fourier transform to $L 1 (G)$ or $L 2 (G)$ , where G is an LCA group. If $G$ is compact, one also obtains a Fourier series, which converges similarly to the $[ - π,π]$ case, but if $G$ is noncompact, one obtains instead a Fourier integral. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and This generalization yields the usual Fourier transform when the underlying locally compact Abelian group is $\mathbb{R}$ . This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and

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 $\sum_{-\infty}^\infty$ by a finite one,

$S_N(x) = \sum_{n=-N}^N \hat{f}(n) e^{inx}.$

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

Least squares property

We say that p is a trigonometric polynomial of degree N when it is of the form

$p(x)=\sum_{n=-N}^N p_n e^{inx}.$

Note that $S N (x)$ is a trigonometric polynomial of degree N. In the Mathematical subfields of Numerical analysis and Mathematical analysis, a trigonometric polynomial is a finite Linear combination of sin( Parseval's theorem implies that

THEOREM. $S N (x)$ is the unique best trigonometric polynomial of degree $N$ approximating $f (x)$ , in the sense that, for any trigonometric polynomial $p\neq S_N$ of degree $N$ , we have $\|S_N - f\| \lneqq \|p - f\|$ .

Here, the Hilbert space norm is

$\| g \| = \sqrt{{1 \over 2\pi} \int_{-\pi}^{\pi} |g(x)|^2 \, dx}.$

Convergence

Main article: Convergence of Fourier series

Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result. In Mathematics, the question of whether the Fourier series of a Periodic function converges to the given function is researched by

THEOREM. If $f\in L^2([-\pi,\pi])$ , then the Fourier series converges in $L 2 ([ - π,π])$ , i. e. , $\|S_N - f\|$ converges to 0 as $N$ goes to infinity.

We have already mentioned that if f is twice continuously differentiable, then $n^2 \hat{f}(n)$ converges to zero as n goes to infinity. This immediately gives a second convergence result.

THEOREM. If $f \in C^2(\mathbb{T})$ , then $\sup_x |f(x) - S_N(x)| \leq \sum_{|n|>N} |\hat{f}(n)|$ converges to zero, i. e. , $S N$ converges to $f$ uniformly. In the mathematical field of analysis, uniform convergence is a type of Convergence stronger than Pointwise convergence.

In particular, $S N$ converges to f pointwise. In Mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function

Many further cases are discussed in the main article, Convergence of Fourier series, ranging from the moderately simple result that the series converges at x if f is differentiable at x, to Lennart Carleson's much more sophisticated result that the Fourier series of an $L 2$ function actually converges almost everywhere. In Mathematics, the question of whether the Fourier series of a Periodic function converges to the given function is researched by Lennart Axel Edvard Carleson (born March 18, 1928) is a Swedish mathematician known as a leader in the field of Harmonic analysis. In Measure theory (a branch of Mathematical analysis) one says that a property holds almost everywhere if the set of elements for which the property does

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.

In 1922, Andrey Kolmogorov 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. Andrey Nikolaevich Kolmogorov (Андрей Николаевич Колмогоров ( April 25, 1903 - October 20, 1987) was a Soviet

Notes

^ Gallica - Fourier, Jean-Baptiste-Joseph (1768-1830). Oeuvres de Fourier. 1888
^ Formally, the inverse transform is given by:
$\begin{align}\mathcal{F}^{-1}\{G(f)\} &=\mathcal{F}^{-1}\left\{ \sum_{n=-\infty}^{\infty} G[n]\cdot \delta \left(f-\frac{n}{\tau}\right)\right\}\\&= \sum_{n=-\infty}^{\infty} G[n]\cdot \mathcal{F}^{-1}\left\{\delta\left(f-\frac{n}{\tau}\right)\right\}\\&= \sum_{n=-\infty}^{\infty} G[n]\cdot e^{i2\pi \frac{n}{\tau} x}\cdot \mathcal{F}^{-1}\{\delta (f)\}\\&= \sum_{n=-\infty}^{\infty} G[n]\cdot e^{i2\pi \frac{n}{\tau} x} \quad = \ \ g(x)\end{align}$
^ 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 Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions In this sense $\mathcal{F}\{e^{i2\pi \frac{n}{\tau} x}\}$ is a Dirac delta function, which is an example of a distribution. The Dirac delta or Dirac's delta is a mathematical construct introduced by the British theoretical physicist Paul Dirac. In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions

References

Joseph Fourier, translated by Alexander Freeman (published 1822, translated 1878, re-released 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.
Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition. Dover Publications, Inc. , New York, 1976. ISBN 0-486-63331-4
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.
Walter Rudin, Principles of mathematical analysis, Third edition. McGraw-Hill, Inc. , New York, 1976. ISBN 0-07-054235-X
William E. Boyce and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems, Eighth edition. John Wiley & Sons, Inc. , New Jersey, 2005. ISBN 0-471-43338-1

External links

Phasor Phactory Allows custom control of the harmonic amplitudes for arbitrary terms
Java applet shows Fourier series expansion of an arbitrary function
Example problems - Examples of computing Fourier Series
Fourier series explanation - A simple, non-mathematical approach
Eric W. Weisstein, Fourier Series at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Fourier Series Module by John H. Mathews
Joseph Fourier - A site on Fourier's life which was used for the historical section of this article

This article incorporates material from example of Fourier series on PlanetMath, which is licensed under the GFDL. PlanetMath is a free, collaborative online Mathematics Encyclopedia.

Dictionary

-noun

(mathematics) A series involving cosine and sine waves.

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