Fourier series

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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.

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.

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.

Definition[edit]

General form[edit]

Given a complex-valued function f of real argument t, f: RC, 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,

 \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 integration bounds.

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 = n\frac{2\pi}{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,

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

where:

For a formal justification, see Modern derivation of the Fourier coefficients below.

Canonical form[edit]

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 non-negative 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[edit]

The form for period T can be easily derived from the canonical one with the change of variable defined by x=\frac{2\pi}{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[edit]

Simple Fourier series[edit]

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.

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.

\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} 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) \\
&{}= 2\frac{(-1)^{n+1}}{n}.\end{align}

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:

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

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

The wave equation[edit]

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[edit]

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)+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.

From time t to time t+dt, 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^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^{inx} 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 multiplicative characters of the group \mathbb{R}/2\pi\mathbb{Z}.

Historical development[edit]

Context[edit]

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 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[edit]

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''\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.

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 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.

The birth of harmonic analysis[edit]

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.

Modern derivation of the Fourier coefficients[edit]

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:RC 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 2π. 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 2π, [-π,π], say.

We will take the functions f:RC 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 \, d\mu=\frac{1}{2\pi}\int_{-\pi}^\pi f(x)\,dx.

Complex Fourier coefficients[edit]

We can make L2(μ) 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^{i n t},n\in\mathbb{Z}\} is an orthonormal basis of L2(μ), which means we can write

f(x)=\sum_{n\in\mathbb{Z}}\left\langle f,e^{i n x}\right\rangle e^{i n x}.

We usually define \forall n\in\mathbb{Z}, c_n=\left\langle f,e^{i n x}\right\rangle. These numbers are called complex Fourier coefficients. Their expression is

c_n = \frac{1}{2\pi} \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[edit]

The sum in the previous section is symmetrical around 0: indeed, except for n=0, a c-n coefficient corresponds to every cn 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_{n\in\mathbb{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 cn 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(n t\right)\, dt\right)\cos\left(n x\right)+\frac{1}{\pi}\left(\int_{-\pi}^\pi f(t)\sin\left(n t\right)\, dt\right)\sin\left(n x\right)\right].

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

a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos\left(n x\right)\, dx,
b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \sin\left(n x\right)\, dx,

we get:

f(x)=\frac{a_0}{2}+\sum_{n>0}\left[a_n\cos\left(n x\right)+b_n\sin\left(n x\right)\right].

Properties[edit]

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)\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 kth derivative of f.

  • For any positive integer k, if f is Ck − 1 and piecewise Ck, then
\lim_{n\rightarrow +\infty}|n^kc_n(f)|=0 because n^kc_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[edit]

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:G\rightarrow\mathbb{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 \langle\cdot,\cdot\rangle on C[G] by: \langle\chi_1, \chi_2\rangle=\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 :GC. The Fourier coefficients of f are defined by: \widehat{f}(\chi)=\langle f,\chi\rangle and we have  f(g) = \int_{\widehat{G}} \widehat{f} (\chi)\chi(g)\,d\chi. If the group is discrete, then the integral reduces to an ordinary sum.

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

\widehat{G}=\{\chi_n:t\mapsto e^{i n t}, n\in\mathbb{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[edit]

Definition of a Fourier series[edit]

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}^Nc_n \chi_n(x) the N-th partial sum of the Fourier series of this function.

Approximation with the partial sums[edit]

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},\dots,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.

This 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 Nth partial sum of the Fourier series. An illustration of this is given on the animated plot of example 1.

Convergence[edit]

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. The proof of this result is simple, unlike Lennart Carleson's much stronger result that the series actually converges almost everywhere.

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 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 theorems[edit]

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[edit]

References[edit]

  • 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[edit]