NationMaster - Encyclopedia: Fourier series

The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. The weights, or coefficients, of the modes, are a one-to-one mapping of the original function. Generalizations include generalized Fourier series and other expansions over orthonormal bases. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... In mathematics, the discrete Fourier transform (DFT) is a transform for Fourier analysis of finite-domain discrete-time signals. ... Given a discrete set of real or complex numbers: (integers), the discrete-time Fourier transform (or DTFT) of is: // Its name implies that the {x[n]} sequence represents the values (aka samples) of a continuous-time function, , at discrete moments in time: , where is the sampling interval (in seconds), and... This is a list of linear transformations of functions related to the Fourier transform. ... Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ... You may be looking for an Injective function, in which (f(a)=f(b)) -> a=b, or a Bijection function, which is both injective and surjective (ie. ... In mathematical analysis, there are many potentially useful generalizations of Fourier series. ... In mathematics, an orthonormal basis of an inner product space V(i. ...

Fourier series serve many useful purposes, as manipulation and conceptualization of the modal coefficients are often easier than with the original function. Areas of application include electrical engineering, vibration analysis, acoustics, optics, signal and image processing, and data compression. Using the tools and techniques of spectroscopy, for example, astronomers can deduce the chemical composition of a star by analyzing the frequency components, or spectrum, of the star's emitted light. Similarly, engineers can optimize the design of a telecommunications system using information about the spectral components of the data signal that the system will carry. See also spectrum analyzer. Electrical Engineers design power systemsâ€¦ â€¦ and complex electronic circuits. ... Oscillation is the variation, typically in time, of some measure as seen, for example, in a swinging pendulum. ... Acoustics is a branch of physics and is the study of sound (mechanical waves in gases, liquids, and solids). ... Table of Opticks, 1728 Cyclopaedia Optics ( appearance or look in ancient Greek) is a branch of physics that describes the behavior and properties of light and the interaction of light with matter. ... Signal processing is the processing, amplification and interpretation of signals, and deals with the analysis and manipulation of signals. ... UPIICSA IPN - Binary image Image processing is any form of information processing for which the input is an image, such as photographs or frames of video; the output is not necessarily an image, but can be for instance a set of features of the image. ... Wavelet compression is a form of data compression well suited for image compression (sometimes also video compression and audio compression). ... Extremely high resolution spectrum of the Sun showing thousands of elemental absorption lines (fraunhofer lines) Spectroscopy is the study of matter and its properties by investigating light, sound, or particles that are emitted, absorbed or scattered by the matter under investigation. ... In most modern usages of the word spectrum, there is a unifying theme of between extremes at either end. ... A spectrum analyzer is a device used to examine the spectral composition of some electrical, acoustic, or optical waveform. ...

The Fourier series is named after the French scientist and mathematician Joseph Fourier, who used them in his influential work on heat conduction, Théorie Analytique de la Chaleur (The Analytical Theory of Heat), published in 1822. Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... Heat conduction or Thermal conduction is the spontaneous transfer of thermal energy through matter, from a region of higher temperature to a region of lower temperature, and hence acts to even out temperature differences. ...

Definition

General form

Given a complex-valued function f of real argument t, f: R → C, where f(t) is piecewise smooth and continuous, periodic with period T, and square-integrable over the interval from $t 1$ to $t 2$ of length T, that is, In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ...

$int_{t_1}^{t_2} |f(t)|^2 dt<+infty$

where

$T = t 2 - t 1$ is the period,
$t 1$ and $t 2$ are times to integrate between.

The Fourier series expansion of f is:

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

where, for any non-negative integer n:

$omega_n = nfrac{2pi}{T}$ is the nth harmonic (in radians) of the function f,
$a_n = frac{2}{T}int_{t_1}^{t_2} f(t) cos(omega_n t), dt$ are the even Fourier coefficients of f, and
$b_n = frac{2}{T}int_{t_1}^{t_2} f(t) sin(omega_n t), dt$ are the odd Fourier coefficients of f.

Equivalently, in complex exponential form, In acoustics and telecommunication, the harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency. ... In mathematics and physics, the radian is a unit of angle measure. ... In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives. ... In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives. ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ...

$f(t) = sum_{n=-infty}^{+infty} c_n e^{i omega_n t}$

where:

$c_n = frac{1}{T}int_{t_1}^{t_2} f(t) e^{-i omega_n t}, dt,$
$i$ is the imaginary unit, and
$e^{i omega_n t} = cos(omega_n t) + i sin(omega_n t)$ in accordance with Euler's formula.

For a formal justification, see Modern derivation of the Fourier coefficients below. In mathematics, the imaginary unit (sometimes also represented by the Latin or the Greek iota) allows the real number system to be extended to the complex number system . ... Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... The Fourier series is a mathematical tool used for analyzing periodic functions by decomposing such a function into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...

Canonical form

In the special case where the period T = 2π, we have

$omega_n = n ,$

In this case, the Fourier series expansion reduces to a particularly simple form:

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

where

$a_n = frac{1}{pi}int_{-pi}^{pi} f(t) cos(nt), dt$
$b_n = frac{1}{pi}int_{-pi}^{pi} f(t) sin(nt), dt$

for any integer n.

or, equivalently:

$f(t) = sum_{n=-infty}^{+infty} c_n e^{i nt}$

where

$c_n = frac{1}{2 pi}int_{-pi}^{pi} f(t) e^{-i nt}, dt = frac{1}{2}(a_n-ib_n).$

Choice of the form

The form for period T can be easily derived from the canonical one with the change of variable defined by $x=frac{2pi}{T}t$ . Therefore, both formulations are equivalent. However, the form for period T is used in most practical cases because it is directly applicable. For the theory, the canonical form is preferred because it is more elegant and easier to interpret mathematically, as will later be seen.

Examples

Simple Fourier series

Let f be periodic of period $2π$ , with $f (x) = x$ for x from −π to π. Note that this function is a periodic version of the identity function. An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

Plot of a periodic identity function - a sawtooth wave.

Animated plot of the first five successive partial Fourier series

We will compute the Fourier coefficients for this function. Image File history File links Periodic_identity. ... Image File history File links Periodic_identity. ... This article or section does not cite its references or sources. ... Image File history File links Periodic_identity_function. ... Image File history File links Periodic_identity_function. ...

$begin{align} a_n &{}= frac{1}{pi}int_{-pi}^{pi}f(x)cos(nx),dx &{}= frac{1}{pi}int_{-pi}^{pi}x cos(nx),dx &{}= 0. end{align}$

$begin{align} b_n &{}= frac{1}{pi}int_{-pi}^{pi}f(x)sin(nx),dx &{}= frac{1}{pi}int_{-pi}^{pi} x sin(nx), dx &{}= frac{2}{pi}int_{0}^{pi} xsin(nx), dx &{}= frac{2}{pi} left(left[-frac{xcos(nx)}{n}right]_0^{pi} + left[frac{sin(nx)}{n^2}right]_0^{pi}right) &{}= 2frac{(-1)^{n+1}}{n}.end{align}$

Notice that a_n are 0 because the $xmapsto xcos(nx)$ are odd functions. Hence the Fourier series for this function is:

$f(x)=frac{a_0}{2} + sum_{n=1}^{infty}left[a_ncosleft(nxright)+b_nsinleft(nxright)right]$

$=2sum_{n=1}^{infty}frac{(-1)^{n+1}}{n} sin(nx), quad forall xin [-pi,pi].$

One application of this Fourier series is to compute the value of the Riemann zeta function at s = 2; by Parseval's theorem, we have: In mathematics, the Riemann zeta function, named after German mathematician Bernhard Riemann, is a function of significant importance in number theory, because of its relation to the distribution of prime numbers. ... In mathematics, Parsevals theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. ...

$frac{1}{2pi} int_{-pi}^pi x^2 dx=frac{1}{2}sum_{n>0}left[2frac{(-1)^n}{n}right]^2$

which yields: $sum_{n>0}frac{1}{n^2}=frac{pi^2}{6}$ .

The wave equation

The wave equation governs the motion of a vibrating string, which may be fastened down at its endpoints. The solution of this problem requires the trigonometric expansion of a general function f that vanishes at the endpoints of an interval x=0 and x=L. The Fourier series for such a function takes the form The wave equation is an important partial differential equation that describes a variety of waves, such as sound waves, light waves and water waves. ...

$f(x) = sum_{n=1}^{infty} b_n sin left( frac{npi}{L} x right)$

where

$b_n = frac{2}{L} int_0^L f(x) sin left( frac{npi}{L} xright), dx.$

Vibrations of air in a pipe that is open at one end and closed at the other are also described by the wave equation. Its solution requires expansion of a function that vanishes at x = 0 and whose derivative vanishes at x=L. The Fourier series for such a function takes the form

$f(x) = sum_{n=1}^{infty} b_n sin left( frac{(2n +1)pi}{2L} x right)$

where

$b_n = frac{2}{L} int_0^L f(x) sin left( frac{(2n+1)pi}{2L} xright), dx.$

Interpretation: decomposing a movement in rotations

Movement in the complex plane

Fourier series have a kinematic interpretation. Indeed, the function $tmapsto f(t)$ can be seen as the movement of an object on a plane (t would then represent time). Since f is complex-valued, we can write Image File history File links Animated_cardioid. ... Image File history File links Animated_cardioid. ... In physics, kinematics is the branch of classical mechanics concerned with describing the motions of objects without considering the factors that cause or affect the motion. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ...

$f(t)=u(t)+i v(t). ,$

for real-valued functions u and v. In this form, we can interpret f as a sum of horizontal and vertical translations. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

From time $t$ to time $t + d t$ , where dt is a very small incremental period, the object moves from the point $A=left[begin{matrix}u(t)v(t)end{matrix}right]$ to the point $B=left[begin{matrix}u(t+dt)v(t+dt)end{matrix}right]$ , which corresponds to an infinitesimal translation in space by the vector $overrightarrow{AB}=left[begin{matrix}u(t+dt)-u(t)v(t+dt)-v(t)end{matrix}right]$ . As a result, we can write f as:

$f(t)=left[begin{matrix}u(dt)-u(0)v(dt)-v(0)end{matrix}right]+left[begin{matrix}u(2dt)-u(dt)v(2dt)-v(dt)end{matrix}right]+cdots+left[begin{matrix}u(t+dt)-u(t)v(t+dt)-v(t)end{matrix}right]$

$=int_0^tfrac{1}{dx}left[begin{matrix}u(x+dx)-u(x)v(x+dx)-v(x)end{matrix}right],dx.$

Now instead of seeing f as a sum of infinitesimal translations, we can see it as an infinite sum of rotations of different radii. This interpretation is convenient, in particular when the movement is periodic.

Let $χ n = e i n x$ be the n-turn per second rotation (of radius 1) (sometimes called character). We want to write f as $f(x)=sum c_n chi_n$ . We can prove (see mathematical derivation below) that the radii of the rotations (the coefficients $c n$ ) are exactly the ones we gave in the previous paragraph.

For example, the plot of the function $f:tmapsto 2cosleft(frac{t}{2}right)e^{frac{3}{2}it}$ is closed, which means the function is periodic. The loop in the curve suggests that it is the sum of two periodic functions, one having a shorter period than the other. Indeed, it can be written: $f (t) = e i t + e 2 i t = χ 1 (t) + χ 2 (t)$ . All its Fourier coefficients are zero except $c 1 = 1$ and $c 2 = 1$ . The graphical interpretation of a rotation is much harder to do than that of the translations because instead of visually seeing the movement from one point to another we have to add the whole motion for the decomposition to make sense (we are reasoning in rotation frequencies rather than in time).

Mathematically, adopting this point of view is seeing Fourier series as a tool to understand linear operators that commute with translations. The functions $χ n$ are precisely the multiplicative characters of the group $mathbb{R}/2pimathbb{Z}$ . In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

Historical development

Context

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. 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. Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ... Madhavan (à´®à´¾à´§à´µà´¨àµ) of Sangamagramam (1350â€“1425) was a prominent mathematician-astronomer from Kerala, India. ... Nilakantha Somayaji (à¤¨à¥€à¤²à¤•à¤£à¥à¤ à¤¸à¥‹à¤®à¤¯à¤¾à¤œà¤¿) (1444-1544), from Kerala, was a major mathematician and astronomer. ... Jyesthadeva (1500-1575), born in Kerala, was a major mathematician, and author of the 1501 Yukti-bhasa, which was a survey of Kerala mathematics and astronomy that was unique at the time for its exacting proofs of the theorems it presented. ... Leonhard Euler (pronounced Oiler; IPA ) (April 15, 1707 â€“ September 7, 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. ... Jean le Rond dAlembert, pastel by Maurice Quentin de La Tour Jean le Rond dAlembert (November 16, 1717 â€“ October 29, 1783) was a French mathematician, mechanician, physicist and philosopher. ... Daniel Bernoulli Daniel Bernoulli (Groningen, February 8, 1700 â€“ Basel, March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland. ... The heat equation is an important partial differential equation which describes the variation of 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 wondered if a function defined on two intervals with two different formulas was still a function). Later, Dirichlet and Riemann expressed Fourier's results with greater precision and formality. Partial plot of a function f. ... In calculus, the integral of a function is an extension of the concept of a sum. ... Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ... Bernhard Riemann. ...

A revolutionary article

In Fourier's work entitled Mémoire sur la propagation de la chaleur dans les corps solides, on pages 218 and 219, we can read the following :

$varphi(y)=acosfrac{pi y}{2}+a'cos 3frac{pi y}{2}+a''cos5frac{pi y}{2}+cdots.$

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

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

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 was the first to recognize that such trigonometric series could represent arbitrary functions, even those with discontinuities. It has required many years to clarify this insight, and it has led to important theories of convergence, function space, and harmonic analysis. Johann Carl Friedrich Gauss or GauÃŸ ( ; Latin: ) (30 April 1777 â€“ 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy, and optics. ... 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 toward a fixed or equilibrium state. ... 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 or a vector space or both. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...

The originality of this work was such that when Fourier submitted his paper in 1807, the committee (composed of no lesser mathematicians than Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour. Joseph-Louis Lagrange, comte de lEmpire (January 25, 1736 â€“ April 10, 1813; b. ... Pierre-Simon Laplace Pierre-Simon Laplace (March 23, 1749 – March 5, 1827) was a French mathematician and astronomer, the discoverer of the Laplace transform and Laplaces equation. ... Etienne-Louis Malus (July 23, 1775 - February 24, 1812) was a French officer, engineer, physicist, and mathematician. ... Adrien-Marie Legendre (September 18, 1752–January 10, 1833) was a French mathematician. ...

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 mathematically equivalent (and correct), 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. In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

Many other Fourier-related transforms have since been defined, extending to other applications the initial idea of representing any periodic function as a superposition of harmonics. This general area of inquiry is now sometimes called harmonic analysis. This is a list of linear transformations of functions related to the Fourier transform. ... The term superposition can have several meanings: Quantum superposition Law of superposition in geology and archaeology Superposition principle for vector fields Superposition Calculus is used for equational first-order reasoning This is a disambiguation page — a navigational aid which lists other pages that might otherwise share the same title. ... Harmonic analysis is the branch of mathematics which studies the representation of functions or signals as the superposition of basic waves. ...

Modern derivation of the Fourier coefficients

The method used by Fourier to derive the coefficients of the series is very practical and well-suited to the problem he was dealing with (heat propagation). However, this method has since been generalized to a much wider class of problems: writing a function as a sum of periodic functions.

More precisely, if f:R → C is a function, we would like to write this function as a sum of trigonometric functions, i.e. $f(x)=sum c_n e^{inx}$ . We have to restrict our choice of functions in order for this to make sense. First of all, if f has period T, then by changing variables, can study $xmapsto fleft(frac{T}{2pi}xright)$ which has period 2π. This simplifies notations a lot and allows us to use a canonical (standard) form. We can restrict the study of $xmapsto fleft(frac{T}{2pi}xright)$ to any interval of length 2π, [-π,π], say.

We will take the functions f:R → C in the set of piecewise continuous, 2π periodic functions with $int_{-pi}^pi |f(x)|^2 , dx<+infty$ . Technically speaking, we are in fact taking functions from the Lp space L²(μ), where μ is the normalized Lebesgue measure of the interval [-π,π] (i.e. such that $int_{[-pi,pi]}f , dmu=frac{1}{2pi}int_{-pi}^pi f(x),dx$ . In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...

Complex Fourier coefficients

We can make L²(μ) into a Hilbert space, which is well-suited for orthogonal projections, by defining the scalar product: The mathematical concept of a Hilbert space (named after the German mathematician David Hilbert) generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the plane and three-dimensional space to spaces of functions. ... In geometry, an orthogonal projection of a k-dimensional object onto a d-dimensional hyperplane (d < k) is obtained by intersections of (k − d)- dimensional hyperplanes drawn through the points of an object orthogonally to the d-hyperplane. ... In mathematics, the dot product (also known as the scalar product and the inner product) is a function (·) : V × V → F, where V is a vector space and F its underlying field. ...

$langle f, g rangle = int_{[-pi,pi]} f overline{g} ,dmu=frac{1}{2pi}int_{-pi}^pi f(x)overline{g(x)},dx,$

where $overline{f(x)}$ denotes the conjugate of f(x). We will denote by $| cdot |$ the associated norm. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ... In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

$E={tmapsto e^{i n t},ninmathbb{Z}}$ is an orthonormal basis of L²(μ), which means we can write In mathematics, an orthonormal basis of an inner product space V(i. ...

$f(x)=sum_{ninmathbb{Z}}leftlangle f,e^{i n x}rightrangle e^{i n x}.$

We usually define $forall ninmathbb{Z}, c_n=leftlangle f,e^{i n x}rightrangle$ . These numbers are called complex Fourier coefficients. Their expression is

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

An equivalent formulation is to write f as a sum of sine and cosine functions.

Real Fourier coefficients

The sum in the previous section is symmetrical around 0: indeed, except for n=0, a c_-n coefficient corresponds to every c_n coefficient. This reminds one of the formulae

$cos(x)=frac{e^{ix}+e^{-ix}}{2}{rm~~~~and~~~~}sin(x)=frac{e^{ix}-e^{-ix}}{2i}.$

It is therefore possible to express the Fourier series with real-valued functions. To do this, we first notice that

$f(x)=sum_{ninmathbb{Z}}c_n e^{i n x}=c_0+sum_{n>0}left[c_{-n}e^{-i n x}+c_n e^{i n x}right].$

After replacing c_n by its expression and simplifying the result we get

$f(x)=c_0+sum_{n>0}left[frac{1}{pi}left(int_{-pi}^pi f(t)cosleft(n tright), dtright)cosleft(n xright)+frac{1}{pi}left(int_{-pi}^pi f(t)sinleft(n tright), dtright)sinleft(n xright)right].$

If, for a non-negative integer n, we define the real Fourier coefficients a_n and b_n by

$a_n = frac{1}{pi}int_{-pi}^{pi} f(x) cosleft(n xright), dx,$

$b_n = frac{1}{pi}int_{-pi}^{pi} f(x) sinleft(n xright), dx,$

we get:

$f(x)=frac{a_0}{2}+sum_{n>0}left[a_ncosleft(n xright)+b_nsinleft(n xright)right].$

Properties

The following properties can be easily derived from Euler's formula: $e^{ix} = cos x + isin x. !$

$a_n=c_n+c_{-n}mbox{ and }b_n=i(c_n-c_{-n})mbox{ for all } n mbox{ and },$

$c_n=frac{a_n-ib_n}{2} mbox{ and }c_{-n}=frac{a_n+ib_n}{2}mbox{ for all } n.$

If f is an odd function, then $a n = 0$ for all $n$ because $f(x)cosleft(npifrac{x}{T}right)$ is then also odd, so its integral on $[ - T, T]$ is zero. If f is an even function, then $b n = 0$ for a similar reason.

If f is piecewise continuous, $lim_{nrightarrow +infty}c_n(f)=0$ , $lim_{nrightarrow +infty}c_{-n}(f)=0$ , $lim_{nrightarrow +infty}a_n(f)=0$ and $lim_{nrightarrow +infty}b_n(f)=0.$

If f is k-times piecewise continuously differentiable, then we can easily compute the Fourier coefficients of $f (k)$ given those of f:

$c_nleft(f^{(k)}right)=(in)^k c_n(f),$

where $f (k)$ denotes the kth derivative of f. Eulers formula, named after Leonhard Euler, is a mathematical formula in complex analysis that shows a deep relationship between the trigonometric functions and the complex exponential function. ... In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives. ... In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking negatives. ... In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ...

For any positive integer k, if f is C^k − 1 and piecewise C^k, then

$lim_{nrightarrow +infty}|n^kc_n(f)|=0$ because $n^kc_n(f)=i^{-k}c_nleft(f^{(k)}right)rightarrow 0.$

This means that the sequence $c n (f)$ is rapidly decreasing. In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ... In mathematics, a function on a normed vector space is said to vanish at infinity if as For example, the function defined on the real line vanishes at infinity. ...

General case

Fourier series take advantage of the periodicity of a function f but what if f is periodic in more than one variable, or for that matter, what if f is not periodic? These problems led mathematicians and theoretical physicists to try to define Fourier series on any group G. The advantage of this is that it allows us, for example, to define Fourier series for functions of several variables. Fourier series and Fourier transforms usually used in signal processing then become special cases of this theory and are easier to interpret.

If G is a locally compact Abelian group and T is the unit circle, we can define the dual of G by $widehat{G} = {chi:Grightarrowmathbb{T} mbox{ homomorphism}}$ . This is the set of rotations on the unit circle and its elements are called characters. We can define a scalar product $langlecdot,cdotrangle$ on C[G] by: $langlechi_1, chi_2rangle=int_{G}chi_1(g) overline{chi_2(g)},dg$ . $widehat{G}$ is then an orthonormal basis of C[G] with respect to this scalar product. Let f :G → C. The Fourier coefficients of f are defined by: $widehat{f}(chi)=langle f,chirangle$ and we have $f(g) = int_{widehat{G}} widehat{f} (chi)chi(g),dchi$ . If the group is discrete, then the integral reduces to an ordinary sum. In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ...

For example, the Fourier series of this article are obtained by taking G=R/2πZ. We get

$widehat{G}={chi_n:tmapsto e^{i n t}, ninmathbb{Z}}$

and

$c_n(f) = widehat{f}(chi_n) = int_G f(g)overline{chi(g)},dg = frac{1}{2 pi}int_{-pi}^{pi} f(t) e^{-i nt},dt.$

Periodic functions in n dimensions can be defined on an n-dimensional torus (the function taking a value at each point on the torus). Such a torus is defined by Tⁿ=Rⁿ/(2πZ)ⁿ. For n=1 we get a circle, for n=2 the cartesian product of two circles, i.e. a torus in the usual sense. Choosing G=Tⁿ gives the corresponding Fourier series.

Approximation and convergence of Fourier series

Definition of a Fourier series

Let $chi_n(x)=e^{inpi frac{x}{T}}$ . We call Fourier series of the function f the series $sum c_n chi_n$ . For any positive integer N, we call $f_N(x)=sum_{n=-N}^Nc_n chi_n(x)$ the N-th partial sum of the Fourier series of this function.

Approximation with the partial sums

Say we want to find the best approximation of f using only the functions $χ n$ for n from $- N$ to N. Let $mathcal{T}_N=left{p=sum_{n=-N}^N x_n chi_n, x_ninmathbb{C}right}$ . We are trying to find coefficients $(x_{-N},dots,x_{N})$ such that $|f-p|$ is minimum (where $| cdot |$ denotes the norm). In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

We have $|f-p|^2=|f|^2-2mbox{Re}langle f,prangle+|p|^2$ , where Re(z) denotes the real part of z.

$langle f,prangle=sum_{n=-N}^Noverline{x_n}langle f,chi_nrangle.$

Parseval's theorem (which can be derived independently from Fourier series) gives us In mathematics, Parsevals theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. ...

$|p|^2=sum_{n=-N}^N|x_n|^2.$

By definition, $c_n=langle f,chi_nrangle$ ; therefore

$|f-p|^2=|f|^2+sum_{n=-N}^Nleft[|c_n-x_n|^2-|c_n|^2right].$

It is clear that this expression is minimum for $x n = c n$ and for this value only.

This means that there is one and only one $f_Ninmathcal{T}_N$ such that

$|f-f_N|=min_{pinmathcal{T}_N}left{|f-p|,pinmathcal{T}_Nright},$

it is given by

$f_N(x)=sum_{n=-N}^N c_n chi_n(x),$

where

$c_n=frac{1}{2T}int_{-T}^T f(t)chi_{-n}(t),dt.$

This means that the best approximation of f we can make using only the functions $chi_n(x)=e^{inpi frac{x}{T}}$ for n from $- N$ to N is precisely the Nth partial sum of the Fourier series. An illustration of this is given on the animated plot of example 1.

Convergence

Main article: Convergence of Fourier series

While the Fourier coefficients a_n and b_n can be formally defined for any function for which the integrals make sense, whether the series so defined actually converges to f(x) depends on the properties of f. In mathematics, the question whether the Fourier series of a periodic function converges to the given function and in what sense is a rich field of research, sometimes called classic harmonic analysis, a branch of pure mathematics. ...

The simplest answer is that if f is square-integrable then In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ...

$lim_{Nrightarrowinfty}int_{-pi}^pileft|f(x)-sum_{n=-N}^{N} c_n,chi_n(x)right|^2,dx=0.$

This is convergence in the norm of the space L². The proof of this result is simple, unlike Lennart Carleson's much stronger result that the series actually converges almost everywhere. In mathematics, the Lp and spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ... Lennart Carleson (b. ...

There are many known tests that ensure that the series converges at a given point x, for example, if the function is differentiable at x. Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at x, then the Fourier series will converge to the average of the left and right limits (but see Gibbs phenomenon). However, a fact that many find surprising, is that the Fourier series of a continuous function need not converge pointwise. In mathematics, the derivative of a function is one of the two central concepts of calculus. ... Approximation of square wave in 5 steps Approximation of square wave in 25 steps Approximation of square wave in 125 steps In mathematics, the Gibbs phenomenon, named after the American physicist J. Willard Gibbs, (also known as ringing artifacts) is the peculiar manner in which the Fourier series of a...

This unpleasant situation is counter-balanced by a theorem by Dirichlet which states that if f is $2 T$ -periodic and piecewise continuously differentiable function, then its Fourier series converges pointwise and $sum_{ninmathbb{Z}} c_n chi_n(x)=frac{f(x^+)+f(x^-)}{2}$ , where $f(x^+)=lim_{trightarrow x, t>x} f(x)$ and $f(x^-)=lim_{trightarrow x, t<x} f(x)$ . If f is continuous as well as piecewise continuously differentiable, then the Fourier series converges uniformly. In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ... Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers if that makes anyone feel good). ...

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. This function is not in $L 2 (μ)$ . Andrey Kolmogorov Andrey Nikolaevich Kolmogorov (ÐÐ½Ð´Ñ€ÐµÌÐ¹ ÐÐ¸ÐºÐ¾Ð»Ð°ÌÐµÐ²Ð¸Ñ‡ ÐšÐ¾Ð»Ð¼Ð¾Ð³Ð¾ÌÑ€Ð¾Ð²) (kahl-mah-GAW-raff) (April 25, 1903 in Tambov - October 20, 1987 in Moscow) was a Soviet mathematician who made major advances in the fields of probability theory and topology. ...

Plancherel's and Parseval's theorems

Another important property of the Fourier series is the Plancherel theorem. Let $f,gin L^2(mu)$ and $c n (f), c n (g)$ be the corresponding complex Fourier coefficients. Then In mathematics, the Plancherel theorem is a result in harmonic analysis, first proved by Michel Plancherel. ...

$sum_{ninmathbb{Z}} c_n(f)overline{c_n(g)} = frac{1}{2T} int_{-T}^T f(x)overline{g(x)},dx$

where $overline{z}$ denotes the conjugate of z. In mathematics, the complex conjugate of a complex number is given by changing the sign of the imaginary part. ...

Parseval's theorem, a special case of the Plancherel theorem, states that: In mathematics, Parsevals theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. ... In mathematics, the Plancherel theorem is a result in harmonic analysis, first proved by Michel Plancherel. ...

$sum_{ninmathbb{Z}} |c_n(f)|^2 = frac{1}{2T} int_{-T}^T |f(x)|^2 ,dx$

which can be restated with the real Fourier coefficients:

$frac{a_0^2}{4} + frac{1}{2} sum_{n=1}^infty left( a_n^2 + b_n^2 right) = frac{1}{2T} int_{-T}^T |f(x)|^2, dx.$

These theorems may be proven using the orthogonality relationships. They can be interpreted physically by saying that writing a signal as a Fourier series does not change its energy.

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.
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. MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ... PlanetMath is a free, collaborative, online mathematics encyclopedia. ...

Categories: PlanetMath sourced articles | Fourier series

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Contents

Definition GA_googleFillSlot("encyclopedia_square");

General form

Canonical form

Choice of the form

Examples

Simple Fourier series

The wave equation

Interpretation: decomposing a movement in rotations

Historical development

Context

A revolutionary article

The birth of harmonic analysis

Modern derivation of the Fourier coefficients

Complex Fourier coefficients

Real Fourier coefficients

Properties

General case

Approximation and convergence of Fourier series

Definition of a Fourier series

Approximation with the partial sums

Convergence

Plancherel's and Parseval's theorems

See also

References

External links

Definition