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Fourier transforms
Continuous Fourier transform
Fourier series
Discrete Fourier transform
Discrete-time Fourier transform
Related transforms
The first four Fourier series approximations for a square wave.

In mathematics, a Fourier series decomposes a periodic function or periodic signal into a sum of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate.

The heat equation is a partial differential equation. 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. 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 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, quantum mechanics, econometrics[1], etc.

Fourier series is named in honour of Joseph Fourier (1768-1830), who made important contributions to the study of trigonometric series, after preliminary investigations by 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 his 1807 Mémoire sur la propagation de la chaleur dans les corps solides 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. Later, Dirichlet and Riemann expressed Fourier's results with greater precision and formality.

Contents

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(2k+1)\frac{\pi y}{2}, and then integrating from y = − 1 to y = + 1 yields:

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

—Joseph Fourier, Mémoire sur la propagation de la chaleur dans les corps solides, pp. 218–219.[2]

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. In what sense that is actually true is a somewhat subtle issue and the attempts over many years to clarify this idea have led to important discoveries in the theories of convergence, function spaces, and harmonic analysis.

When Fourier submitted his paper in 1807, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.

Birth of harmonic analysis

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

Many other Fourier-related transforms have since been defined, extending the initial idea to other applications. This general area of inquiry is now sometimes called harmonic analysis. A Fourier series, however, can be used only for periodic functions.

Definition

In this section, ƒ(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 ƒ(x + 2π) = ƒ(x), for all real numbers x. We will attempt to write such a function as an infinite sum, or series of simpler 2π–periodic functions. We will start by using an infinite sum of sine and cosine functions on the interval [−ππ], as Fourier did (see the quote above), and we will then discuss different formulations and generalizations.

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

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

a_n = \frac{1}{p}\int_{-p}^p f(x) \cos(nx)\, dx, \quad n \ge 0

and

b_n = \frac{1}{p}\int_{-p}^p f(x) \sin(nx)\, dx, \quad n \ge 1

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

(S_N f)(x) = \frac{a_0}{2} + \sum_{n=1}^N \, [a_n \cos(nx) + b_n \sin(nx)], \quad N \ge 0.

The partial sums for ƒ are trigonometric polynomials. One expects that the functions SN ƒ approximate the function ƒ, and that the approximation improves as N tends to infinity. The infinite sum

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

is called the Fourier series of ƒ.

The Fourier series does not always converge, and even when it does converge for a specific value x0 of x, the sum of the series at x0 may differ from the value ƒ(x0) of the function. It is one of the main questions in harmonic analysis to decide when Fourier series converge, and when the sum is equal to the original function. If a function is square-integrable on the interval [−ππ], then the Fourier series converges to the function at almost every point. In engineering applications, the Fourier series is generally presumed to converge everywhere except at discontinuities, since the functions encountered in engineering are more well behaved than the ones that mathematicians can provide as counter-examples to this presumption. In particular, the Fourier series converges absolutely and uniformly to ƒ(x) whenever the derivative of ƒ(x) (which may not exist everywhere) is square integrable.[3] See Convergence of Fourier series.

It is possible to define Fourier coefficients for more general functions or distributions, in such cases convergence in norm or weak convergence is usually of interest.

Example: a simple Fourier series

Plot of a periodic identity function—a sawtooth wave
Animated plot of the first five successive partial Fourier series

We now use the formula above to give a Fourier series expansion of a very simple function. Consider a sawtooth wave

f(x) = x, \quad \mathrm{for } -\pi < x < \pi,
f(x + 2\pi) = f(x), \quad \mathrm{for } -\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, \quad n \ge 0. \ b_n &{}= \frac{1}{\pi}\int_{-\pi}^{\pi} x \sin(nx)\, dx = -\frac{2}{n}\cos(n\pi) + \frac{2}{n^2\pi}\sin(n\pi) = 2 \, \frac{(-1)^{n+1}}{n}, \quad n \ge 1.\end{align}

It can be proved that the Fourier series converges to ƒ(x) at every point x where ƒ is differentiable, 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 x - \pi \notin 2 \pi \mathbf{Z}. \end{align}

 

 

 

 

(Eq.1)

   

When x = π, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of ƒ at x = π. This is a particular instance of the Dirichlet theorem for Fourier series.

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 ƒ(x) = x, and so it is not immediately apparent why one would need this Fourier series. While there are many applications, we cite Fourier's motivation of solving the heat equation. For example, consider a metal plate in the shape of a square whose side measures π meters, with coordinates (xy) ∈ [0, π] × [0, π]. If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by y = π, is maintained at the temperature gradient T(xπ) = 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. 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(xy) 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 example generalizes and one may compute ζ(2n), for any positive integer n.

Exponential Fourier series

We can use Euler's formula,

 e^{inx} = \cos(nx)+i\sin(nx), \,

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 an, bn, cn are related via

a_n = { c_n + c_{-n} }\text{ for }n=0,1,2,\dots,\,

and

b_n = i( c_{n} - c_{-n} )\text{ for }n=1,2,\dots\,

The notation cn is inadequate for discussing the Fourier coefficients of several different functions. Therefore it is customarily replaced by a modified form of ƒ (in this case), such as F or \scriptstyle\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 engineering, particularly when the variable x represents time, the coefficient sequence is called a frequency domain representation. Square brackets are often used to emphasize that the domain of this function is a discrete set of frequencies.

Fourier series on a general interval [ab]

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

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

If a function is square-integrable in the interval [aa + τ], it can be represented in that interval by the formula above. 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 τ-periodic, then a is an arbitrary choice. Two popular choices are a = 0, and a = −τ/2.

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

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

where variable ƒ represents a continuous frequency domain. When variable x has units of seconds, ƒ has units of hertz. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of 1/τ, which is called the fundamental frequency. The original g(x) can be recovered from this representation by an inverse Fourier transform:

 \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 \underbrace{\mathcal{F}^{-1}\left\{\delta\left(f-\frac{n}{\tau}\right)\right\}}_{e^{i2\pi \frac{n}{\tau} x}\cdot \underbrace{\mathcal{F}^{-1}\{\delta (f)\}}_1 } \ &= \sum_{n=-\infty}^\infty G[n]\cdot e^{i2\pi \frac{n}{\tau} x} \quad = \ \ g(x). \end{align}

The function G(ƒ) is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent.[4]

Fourier series on a square

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

f(x,y) = \sum_{j,k \in \mathbb{Z}\text{ (integers)}} 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 on the square is in image compression. In particular, the jpeg image compression standard uses the two-dimensional discrete cosine transform, which is a Fourier transform using the cosine basis functions.

Hilbert space interpretation

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 L2([ − π,π]) of square-integrable functions of [ − π,π]. 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 for 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. The version with sines and cosines is also justified with the Hilbert space interpretation. Indeed, the sines and cosines form an orthogonal set:

\int_{-\pi}^{\pi} \cos(mx)\, \cos(nx)\, dx = \pi \delta_{mn}, \quad m, n \ge 1, \,
\int_{-\pi}^{\pi} \sin(mx)\, \sin(nx)\, dx = \pi \delta_{mn}, \quad m, n \ge 1

(where δmn is the Kronecker delta), and

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

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

Properties

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

  • If ƒ is a 2π-periodic odd function, then an = 0  for all n.
  • If ƒ is a 2π-periodic even function, then bn = 0  for all n.
  • 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.
  • Plancherel's theorem. 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.

General case

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

Generalized functions

One can extend the notion of Fourier coefficients to functions which are not square-integrable, and even to objects which are not functions. This is very useful in engineering and applications because we often need to take the Fourier series of a periodic repetition of a Dirac delta function. The Dirac delta δ is not actually a function; still, it has a Fourier transform and its periodic repetition has a Fourier series:

\hat{\delta}(n)={1 \over 2\pi}\text{ for every }n.\,

This generalization to distributions enlarges the domain of definition of the Fourier transform from L2([−ππ]) to a superset of L2. The Fourier series converges weakly.

Compact groups

One of the interesting properties of the Fourier transform which we have mentioned, is that it carries convolutions to pointwise products. If that is the property which we seek to preserve, one can produce Fourier series on any compact group. Typical examples include those classical groups that are compact. This generalizes the Fourier transform to all spaces of the form L2(G), where G is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products. The Fourier series exists and converges in similar ways to the [−ππ] case.

An alternative extension to compact groups is the Peter–Weyl theorem, which proves results about representations of compact groups analogous to those about finite groups.

Riemannian manifolds

The atomic orbitals of chemistry are spherical harmonics and can be used to produce Fourier series on the sphere.

If the domain is not a group, then there is no intrinsically defined convolution. However, if X is a compact Riemannian manifold, it has a Laplace-Beltrami operator. The Laplace-Beltrami operator is the differential operator that corresponds to Laplace operator for the Riemannian manifold X. Then, by analogy, one can consider heat equations on X. Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace-Beltrami operator as a basis. This generalizes Fourier series to spaces of the type L2(X), where X is a Riemannian manifold. The Fourier series converges in ways similar to the [−ππ] 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.

Locally compact Abelian groups

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

This generalizes the Fourier transform to L1(G) or L2(G), where G is an LCA group. If G is compact, one also obtains a Fourier series, which converges similarly to the [−ππ] case, but if G is noncompact, one obtains instead a Fourier integral. This generalization yields the usual Fourier transform when the underlying locally compact Abelian group is \mathbb{R}.

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 f)(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 (SN ƒ)(x) converge to ƒ(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 SN ƒ is a trigonometric polynomial of degree N. Parseval's theorem implies that

Theorem. The trigonometric polynomial SN ƒ is the unique best trigonometric polynomial of degree N approximating ƒ(x), in the sense that, for any trigonometric polynomial p\neq S_N f of degree N, we have  \|S_N f - f\|_2 < \|p - f\|_2.

Here, the Hilbert space norm is

\| g \|_2 = \sqrt{{1 \over 2\pi} \int_{-\pi}^{\pi} |g(x)|^2 \, dx}.

Convergence

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

Theorem. If ƒ belongs to L2([−π, π]), then the Fourier series converges to ƒ in L2([−π, π]), that is,  \|S_N f - f\|_2 converges to 0 as N goes to infinity.

We have already mentioned that if ƒ is continuously differentiable, then  i n \hat{f}(n)  is the nth Fourier coefficient of the derivative ƒ′. It follows, essentially from the Cauchy-Schwarz inequality, that the Fourier series of ƒ is absolutely summable. The sum of this series is a continuous function, equal to ƒ, since the Fourier series converges in the mean to ƒ:

Theorem. If  f \in C^1(\mathbb{T}), then the Fourier series converges to ƒ uniformly (and hence also pointwise.)

This result can be proven easily if ƒ is further assumed to be C2, since in that case n^2\hat{f}(n) tends to zero as n\to\infty. More generally, the Fourier series is absolutely summable, thus converges uniformly to ƒ, provided that ƒ satisfies a Hölder condition of order α > ½. In the absolutely summable case, the inequality  \sup_x |f(x) - (S_N f)(x)| \le \sum_{|n| > N} |\hat{f}(n)|  proves uniform convergence.

Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at x if ƒ is differentiable at x, to Lennart Carleson's much more sophisticated result that the Fourier series of an L2 function actually converges almost everywhere.

These theorems, and informal variations of them that don't specifty the convergence conditions, are sometimes referred to generically as "Fourier's theorem" or "the Fourier theorem".[5][6][7][8]

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. He later constructed an example of an integrable function whose Fourier series diverges everywhere (Katznelson 1976).

See also

Notes

  1. ^ Marc Nerlove, David M. Grether, Jose L. Carvalho, 1995, Analysis of Economic Time Series. Economic Theory, Econometrics, and Mathematical Economics. Elsevier.
  2. ^ Gallica - Fourier, Jean-Baptiste-Joseph (1768-1830). Oeuvres de Fourier. 1888
  3. ^ Georgi P. Tolstov (1976). Fourier Series. Courier-Dover. ISBN 0486633179. http://books.google.com/books?id=XqqNDQeLfAkC&pg=PA82&dq=fourier-series+converges+continuous-function&ei=L0rJSMvANIPsswOs-pzXDA&sig=ACfU3U3teR3Wwlu7HYq_qHV4QZqj6sYP5A. 
  4. ^ 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 this sense \mathcal{F}\{e^{i2\pi \frac{n}{\tau} x}\} is a Dirac delta function, which is an example of a distribution.
  5. ^ William McC. Siebert (1985). Circuits, signals, and systems. MIT Press. p. 402. ISBN 9780262192293. http://books.google.com/books?id=zBTUiIrb2WIC&pg=PA402&dq=%22fourier%27s+theorem%22&lr=&as_brr=3&as_pt=ALLTYPES&ei=UZ3mSeO2A5PikwSRspGSAQ. 
  6. ^ L. Marton and Claire Marton (1990). Advances in Electronics and Electron Physics. Academic Press. p. 369. ISBN 9780120146505. http://books.google.com/books?id=27c1WOjCBX4C&pg=PA369&dq=%22fourier+theorem%22&lr=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_brr=3&as_pt=ALLTYPES&ei=bpXmSde-MI6QkASd_ZWfAQ. 
  7. ^ Hans Kuzmany (1998). Solid-state spectroscopy. Springer. p. 14. ISBN 9783540639138. http://books.google.com/books?id=-laOoZitZS8C&pg=PA14&dq=%22fourier+theorem%22&lr=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_brr=3&as_pt=ALLTYPES&ei=bpXmSde-MI6QkASd_ZWfAQ#PPA14,M1. 
  8. ^ Karl H. Pribram, Kunio Yasue, and Mari Jibu (1991). Brain and perception. Lawrence Erlbaum Associates. p. 26. ISBN 9780898599954. http://books.google.com/books?id=nsD4L2zsK4kC&pg=PA26&dq=%22fourier+theorem%22&lr=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_brr=3&as_pt=ALLTYPES&ei=bpXmSde-MI6QkASd_ZWfAQ#PPA26,M1. 

References

  • 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
  • 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.
  • Katznelson, Yitzhak (1976), An introduction to harmonic analysis (Second corrected ed.), New York: Dover Publications, Inc, 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

External links

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Study guide

Up to date as of January 14, 2010

From Wikiversity

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.

Contents

Definition

General form

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 t1 to t2 of length T, that is,

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

where

  • T = t2t1 is the period,
  • t1 and t2 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:

  • 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.

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 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

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

Simple Fourier series

Let f be periodic of period , 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

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)+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 χn = einx 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 cn) 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) = eit + e2it = χ1(t) + χ2(t). All its Fourier coefficients are zero except c1 = 1 and c2 = 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 χn are precisely the multiplicative 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 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''\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

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

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

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

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

  • The following properties can be easily derived from Euler's formula: e^{ix} = \cos x + i\sin 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 an = 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 bn = 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.
  • If f is k-times piecewise continuously differentiable, then we can easily compute the Fourier coefficients of f(k) given those of f:
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 cn(f) is rapidly decreasing.

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: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/(2π 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. Choosing G=Tn gives the corresponding Fourier series.

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}^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 χn 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 xn = cn 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

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. 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 L2(μ).

Plancherel's and Parseval's theorems

Another important property of the Fourier series is the Plancherel theorem. Let f,g\in L^2(\mu) and cn(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

  • 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


Wiktionary

Up to date as of January 15, 2010

Definition from Wiktionary, a free dictionary

Contents

English

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Wikipedia

Etymology

Named after Joseph Fourier, a French mathematician

Noun

Singular
Fourier series

Plural
Fourier series

Fourier series (plural Fourier series)

  1. (mathematics) A series of cosine and sine functions or complex exponentials resulting from the decomposition of a periodic function.

Translations

  • Dutch: Fourierreeks







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