Fourier series

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Fourier series is a mathematical tool used for analyzing an arbitrary periodic function by decomposing the function into a sum of much simpler sinusoidal component functions, which differ from each other only in amplitude, frequency, and phase. This analysis serves many useful purposes, among which is making it easier to manipulate the original function, both analytically and graphically.

Areas of application include electrical engineering, 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.

Fourier series are named after the French scientist and mathematician Joseph Fourier, who used them in his influential work on heat conduction, The Analytical Theory of Heat, published in 1822.

Definition

General form

Given a complex-valued function f of real argument t, f:{\mathbb  {R}}\rightarrow {\mathbb  {C}}, where f(t) is piecewise continuous, periodic in period T, and square-integrable over the interval from –T/2 to +T/2, or equivalently,

\int _{{-T/2}}^{{T/2}}|f(t)|^{2}dt<+\infty ,

then the Fourier series expansion of f about t = 0 is:

f(t)={\frac  {a_{0}}{2}}+\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}=n{\frac  {2\pi }{T}}

is the nth harmonic (in radians) of the function f, and the Fourier coefficients of f are:

  • a_{n}={\frac  {2}{T}}\int _{{-T/2}}^{{T/2}}f(t)\cos(\omega _{n}t)dt
  • b_{n}={\frac  {2}{T}}\int _{{-T/2}}^{{T/2}}f(t)\sin(\omega _{n}t)dt,

Equivalently, in exponential form,

f(t)=\sum _{{n=-\infty }}^{{+\infty }}c_{n}e^{{i\omega _{n}t}}

where:

c_{n}={\frac  {1}{T}}\int _{{-T/2}}^{{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 Mathematical derivation of the Fourier coefficients below.

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  {a_{0}}{2}}+\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 positive integer n.

or, equivalently:

f(t)=\sum _{{n=-\infty }}^{{+\infty }}c_{n}e^{{int}}

where:

c_{n}={\frac  {1}{2\pi }}\int _{{-\pi }}^{{\pi }}f(t)e^{{-int}}dt


Many mathematicians prefer this form for its simplicity and elegance, whereas scientists and engineers often prefer the more general form, which has important applications in the physical world.

Examples

Simple Fourier series

Let f be periodic of period 2\pi , with f(x)=x for x from -π to π. Note that this function is a periodic version of the identity function.

File:Periodic identity.png
Plot of a periodic identity function
File:Animated.gif
Animated plot of the first five successive partial Fourier series

We will compute the Fourier coefficients for this function.


a_{n}\, ={\frac  {1}{\pi }}\int _{{-\pi }}^{{\pi }}f(x)\cos(nx)\,dx
={\frac  {1}{\pi }}\int _{{-\pi }}^{{\pi }}x\cos(nx)\,dx
a_{n}\, =0\,
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 }}x\sin(nx)\,dx
={\frac  {2}{\pi }}\left(\left[-{\frac  {x\cos(nx)}{n}}\right]_{0}^{{\pi }}+\left[{\frac  {\sin(nx)}{n^{2}}}\right]_{0}^{{\pi }}\right)
b_{n}\, =2{\frac  {(-1)^{{n+1}}}{n}}

Notice that an are 0 because the x\mapsto x\cos(nx) are odd functions. Hence the Fourier series for this function is:

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 \forall x\in [-\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:

\int _{{-\pi }}^{\pi }x^{2}dx={\frac  {1}{2}}\sum _{{n>0}}\left[2{\frac  {(-1)^{n}}{n}}\right]^{2}

which yields: \sum _{{n>1}}{\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

f(x)=\sum _{{n=1}}^{{\infty }}b_{n}\sin \left({\frac  {n\pi }{L}}x\right)

where

b_{n}={\frac  {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac  {n\pi }{L}}x\right)\,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}}x\right)\,dx

Interpretation: decomposing a movement in rotations

File:Animated cardioid.gif
Movement in the complex plane

Fourier series have a kinematic interpretation. indeed, the function t\mapsto 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

f(t)=u(t)+iv(t)\,.

for real-valued functions u and v. In this form, we can interpret f as a sum of horizontal and vertical translations.

From time t to time t+dt, where dt is an 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}^{t}{\frac  {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 \chi _{n}=e^{{ix}} 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:t\mapsto 2\cos \left({\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^{{it}}+e^{{2it}}=\chi _{1}(t)+\chi _{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 \chi _{n} are precisely the mutliplicative characters of the group {\mathbb  {R}}/2\pi {\mathbb  {Z}}.

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.

From a modern point of view, Fourier's results are somewhat informal, due in no small part 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.

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)=a\cos {\frac  {\pi y}{2}}+a'\cos 3{\frac  {\pi y}{2}}+a''\cos 5{\frac  {\pi y}{2}}+....
Multiplying both sides by \cos(2i+1){\frac  {\pi y}{2}}, and then intergrating from y=-1 to y=+1 yields:
a_{i}=\int _{{-1}}^{1}\varphi (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.

The originality of this work was such that when Fourier submitted his paper in 1807, the comittee (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.

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.

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.

Mathematical derivation of the Fourier coefficients

If f:{\mathbb  {R}}\rightarrow {\mathbb  {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 x\mapsto f\left({\frac  {T}{2\pi }}x\right) which has period 2\pi . This simplifies notations a lot and allows us to use a canonical (standard) form. We can restrict the study of x\mapsto f\left({\frac  {T}{2\pi }}x\right) to any interval of length 2\pi , [-\pi ,\pi ], say.

We will take the functions f:{\mathbb  {R}}\rightarrow {\mathbb  {C}} in the set of piecewise continuous, 2\pi periodic functions with \int _{{-\pi }}^{\pi }|f(x)|^{2}dx<+\infty . Technically speaking, we are in fact taking functions from the Lp space L^{2}(\mu ), where \mu is the standardized Lebesgue measure of the interval [-\pi ,\pi ] (i.e. such that \int _{{[-\pi ,\pi ]}}fd\mu ={\frac  {1}{2\pi }}\int _{{-\pi }}^{\pi }f(x)dx).

Complex Fourier coefficients

We can make L^{2}(\mu ) into a Hilbert space, which is well-suited for orthogonal projections, by defining the scalar product:

\langle f,g\rangle =\int _{{[-\pi ,\pi ]}}f\overline {g}d\mu ={\frac  {1}{2\pi }}\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.

E=\{t\mapsto e^{{int}},n\in {\mathbb  {Z}}\} is an orthonormal basis of L^{2}(\mu ), which means we can write:

f(x)=\sum _{{n\in {\mathbb  {Z}}}}\langle f,e^{{int}}\rangle e^{{inx}}

We usually define \forall n\in {\mathbb  {Z}},c_{n}=\langle f,e^{{inx}}\rangle . These numbers are called complex Fourier coefficients. Their expression is:

c_{n}={\frac  {1}{2\pi }}\int _{{-\pi }}^{{\pi }}f(x)e^{{-inx}}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 arround 0: indeed, except for n=0, a c_{{-n}} coefficient corresponds to every c_{n} coefficient. This reminds of the formulae:

\cos(x)={\frac  {e^{{ix}}+e^{{-ix}}}{2}} and \sin(x)={\frac  {e^{{ix}}-e^{{-ix}}}{2i}}

This means we probably can express the sum above with real-valued functions. To do this, we first notice that:

f(x)=\sum _{{n\in {\mathbb  {Z}}}}c_{n}e^{{inx}}=c_{0}+\sum _{{n>0}}\left[c_{{-n}}e^{{-inx}}+c_{n}e^{{inx}}\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)\cos \left(nt\right)\,dt\right)\cos \left(nx\right)+{\frac  {1}{\pi }}\left(\int _{{-\pi }}^{\pi }f(t)\sin \left(nt\right)\,dt\right)\sin \left(nx\right)\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)\cos \left(nx\right)\,dx,
b_{n}={\frac  {1}{\pi }}\int _{{-\pi }}^{{\pi }}f(x)\sin \left(nx\right)\,dx,

we get:

f(x)={\frac  {a_{0}}{2}}+\sum _{{n>0}}\left[a_{n}\cos \left(nx\right)+b_{n}\sin \left(nx\right)\right].

Properties

a_{n}=c_{n}+c_{{-n}}{\mbox{,   }}b_{n}=i(c_{n}-c_{{-n}}){\mbox{ and  }}c_{n}={\frac  {a_{n}-ib_{n}}{2}} for all n.
  • If f is an odd function, then a_{n}=0 for all n because f(x)\cos \left(n\pi {\frac  {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 _{{n\rightarrow +\infty }}c_{n}(f)=0, \lim _{{n\rightarrow +\infty }}c_{{-n}}(f)=0, \lim _{{n\rightarrow +\infty }}a_{n}(f)=0 and \lim _{{n\rightarrow +\infty }}b_{n}(f)=0.
c_{n}\left(f^{{(k)}}\right)=(in)^{k}c_{n}(f),

where f^{{(k)}} denotes the k-th derivative of f.

  • For any positive integer k, if f is Ck-1 and piecewise Ck, then
\lim _{{n\rightarrow +\infty }}|n^{k}c_{n}(f)|=0 because n^{k}c_{n}(f)=i^{{-k}}c_{n}\left(f^{{(k)}}\right)\rightarrow 0.

This means that the sequence c_{n}(f) is rapidly decreasing.

General Case

Fourier series take advantage of the periodicity of a function f. What if, instead of being defined on a circle, f were defined on a torus (i.e. periodic in more than one variable)? An n dimensional torus is defined by {\mathbb  {T}}^{n}={\mathbb  {R}}^{n}/(2\pi {\mathbb  {Z}})^{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.

Let \left\{x\mapsto \chi _{k}(x),k\in {\mathbb  {Z}}^{n}\right\} be an orthonormal basis of L^{2}({\mathbb  {T}}^{n}). Then for any f\in L^{2}({\mathbb  {T}}^{n}), we can write f(x)=\sum _{{k\in {\mathbb  {Z}}^{k}}}{\hat  {f}}_{k}\chi _{k}(x) and the {\hat  {f}}_{k} are called Fourier coefficients.

Approximation and convergence of Fourier series

Definition of a Fourier series

Let \chi _{n}(x)=e^{{in\pi {\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}}^{N}c_{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 \chi _{n} for n from -N to N. Let {\mathcal  {T}}_{N}=\left\{p=\sum _{{n=-N}}^{N}x_{n}\chi _{n},x_{n}\in {\mathbb  {C}}\right\}. We are trying to find coefficients (x_{{-N}},\cdots ,x_{{N}}) such that \|f-p\| is minimum (where \|\cdot \| denotes the norm).

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

\langle f,p\rangle =\sum _{{n=-N}}^{N}\overline {x_{n}}\langle f,\chi _{n}\rangle

Parseval's theorem (which can be derived independently from Fourier series) gives us:

\|p\|^{2}=\sum _{{n=-N}}^{N}|x_{n}|^{2}

By definition, c_{n}=\langle f,\chi _{n}\rangle , therefore:

\|f-p\|^{2}=\|f\|^{2}+\sum _{{n=-N}}^{N}\left[|c_{n}-x_{n}|^{2}-|c_{n}|^{2}\right]

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_{N}\in {\mathcal  {T}}_{N} such that:

\|f-f_{N}\|=\min _{{p\in {\mathcal  {T}}_{N}}}\left\{\|f-p\|,p\in {\mathcal  {T}}_{N}\right\}

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

In layman's terms, it means that the best approximation of f we can make using only the functions \chi _{n}(x)=e^{{in\pi {\frac  {x}{T}}}} for n from -N to N is precisely the N-th 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 an and bn 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.

The simplest answer is that if f is square-integrable then

\lim _{{N\rightarrow \infty }}\int _{{-\pi }}^{\pi }\left|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, which means that the series converges almost everywhere to f.

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.

This unpleasant situation is counter-balanced by a theorem by Dirichlet which states that if f is 2T-periodic and piecewise continuously differentiable function, then its Fourier series converges pointwise and \sum _{{n\in {\mathbb  {Z}}}}c_{n}\chi _{n}(x)={\frac  {f(x^{+})+f(x^{-})}{2}}, where f(x^{+})=\lim _{{t\rightarrow x,t>x}}f(x) and f(x^{-})=\lim _{{t\rightarrow x,t<x}}f(x). If f is continuous as well as piecewise continuously differentiable, then the Fourier series converges in norm and hence converges uniformly.

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}(\mu ).

Plancherel's and Parseval's theorem

Another important property of the Fourier series is the Plancherel theorem. Let f,g\in L^{2}(\mu ) and c_{n}(f),c_{n}(g) be the corresponding complex Fourier coefficients. Then:

\sum _{{n\in {\mathbb  {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.

Parseval's theorem, a special case of the Plancherel theorem, states that:

\sum _{{n\in {\mathbb  {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.

See also

References

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}}.{{ #if: |  “{{{quote}}}” }}  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 0486633314
  • Felix Klein, Development of mathematics in the 19th century. Mathsci Press Brookline, Mass, 1979. Translated by M. Ackerman from Vorlesungen uber die Entwicklung der Matematik im 19 Jahrhundert, Springer, Berlin, 1928.
  • Walter Rudin, Principles of mathematical analysis, Third edition. McGraw-Hill, Inc., New York, 1976. ISBN 007054235X

External links

  • Java applet shows Fourier series expansion of an arbitrary function
  • 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.