Fourier series

In mathematics, the Fourier series is a type of Fourier analysis, which is used on functions that might otherwise be difficult or impossible to analyze. The series decomposes a periodic function into a sum of simple functions, which may be sines and cosines or may be complex exponentials. Introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate, it led to a revolution in mathematics, forcing mathematicians to reexamine the foundations of mathematics and leading to many modern theories such as integration and Hilbert spaces.

The heat equation is a partial differential equation. Prior to Fourier's work, mathematicians could not give a solution to the heat equation in a general situation, although they could easily give an exact solution if the heat source behaved in a simple way, which can be mathematically described using sine and cosine waves. 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, image processing, etc.

Historical development

Fourier series are named in honor of Joseph Fourier (1768-1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Madhava, Nilakantha Somayaji, Jyesthadeva, Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. 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

to Failed to parse (Missing texvc executable; please see math/README to configure.): y=+1
yields:

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 [...] his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.

The birth of harmonic analysis

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

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.

Definition

In this section, f(x) denotes a function of the real variable x. This function is usually taken to be periodic, of period 2π, which is to say that f(x+2π) = f(x), for all real numbers x. We will show how to write such a function as an infinite sum, or series. We will start by using an infinite sum of sine and cosine functions of the interval [-π,π], as Fourier did (see the quote above), and we will then discuss different formulations and generalizations.

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

Let Failed to parse (Missing texvc executable; please see math/README to configure.): a_0,a_1,...

and Failed to parse (Missing texvc executable; please see math/README to configure.): b_1, b_2, ...
be real or complex numbers, called the Fourier coefficients. The infinite sum

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = \frac{a_0}{2} +\sum_{n=1}^{\infty}[a_n \cos(nx) + b_n \sin(nx)]

is a Fourier series for the interval Failed to parse (Missing texvc executable; please see math/README to configure.): [-\pi,\pi] . The Fourier series does not always converge, but see below. If a function is square-integrable in the interval Failed to parse (Missing texvc executable; please see math/README to configure.): [-\pi,\pi] , then it can be represented in that interval by the previous formula, and the Fourier coefficients are given by

Failed to parse (Missing texvc executable; please see math/README to configure.): a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos(nx)\, dx

and

Failed to parse (Missing texvc executable; please see math/README to configure.): b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \sin(nx)\, dx.

Example: a simple Fourier series

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

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = x, \quad \mathrm{for} \quad -\pi < x < \pi,

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x + 2\pi) = f(x), \quad \mathrm{for} \quad -\infty < x < \infty.

In this case, the Fourier coefficients are given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} a_n &{} = \frac{1}{\pi}\int_{-\pi}^{\pi}x \cos(nx)\,dx = 0. \\ b_n &{}= \frac{1}{\pi}\int_{-\pi}^{\pi} x \sin(nx)\, dx = 2\frac{(-1)^{n+1}}{n}.\end{align}

And therefore:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} f(x) &= \frac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n\cos\left(nx\right)+b_n\sin\left(nx\right)\right] \\ &=2\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} \sin(nx), \quad \mathrm{for} \quad -\infty < x < \infty . \;\;\; (*) \end{align}

One notices that the Fourier series expansion of our function looks much less simple than the formula f(x)=x, and so it is not immediately apparent why one would need this Fourier series. While there are many applications, we cite Fourier's motivation of solving the heat equation. For example, consider a metal plate in the shape of a square whose side measures π meters, with coordinates Failed to parse (Missing texvc executable; please see math/README to configure.): (x,y) \in [0,\pi] \times [0,\pi] . If there is no heat source within the plate, and if three of the four sides are held at 0 degrees celsius, while the fourth side, given by y=π, is maintained at the temperature gradient T(x,π) = x degrees celsius, for x in (0,π), then one can show that the stationary heat distribution (or the heat distribution after a long period of time has elapsed) is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 (*) by sinh(ny)/sinh(nπ). While our example function f(x) seems to have a needlessly complicated Fourier series, the heat distribution T(x,y) is nontrivial. The function T cannot be written as a closed-form expression. This method of solving the heat problem was only made possible by Fourier's work.

Another application of this Fourier series is to solve the Basel problem by using Parseval's theorem. The example generalizes and one may compute zeta(2n), for any positive integer n.

The modern version using complex exponentials

We can use Euler's formula, Failed to parse (Missing texvc executable; please see math/README to configure.): e^{inx}=\cos(nx)+i\sin(nx) , where Failed to parse (Missing texvc executable; please see math/README to configure.): i

is the imaginary unit, to give a more concise formula:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x) = \sum_{n=-\infty}^{\infty} c_n e^{inx}.

The Fourier coefficients are then given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) e^{-inx}\, dx.

The Fourier coefficients Failed to parse (Missing texvc executable; please see math/README to configure.): a_n, b_n, c_n

are related via

Failed to parse (Missing texvc executable; please see math/README to configure.): a_n = { c_n + c_{-n} }

for Failed to parse (Missing texvc executable; please see math/README to configure.): n=0,1,2,\dots,

and

Failed to parse (Missing texvc executable; please see math/README to configure.): b_n = i( c_{n} - c_{-n} )

for Failed to parse (Missing texvc executable; please see math/README to configure.): n=1,2,\dots

The notation Failed to parse (Missing texvc executable; please see math/README to configure.): c_n

is inadequate for discussing the Fourier coefficients of several different functions.    Therefore it is customarily replaced by a modified form of Failed to parse (Missing texvc executable; please see math/README to configure.): f\,
(in this case), such as Failed to parse (Missing texvc executable; please see math/README to configure.): F\,
or Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{f},

  and functional notation often replaces subscripting.  Thus:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} f(x) &= \sum_{n=-\infty}^{\infty} \hat{f}(n)\cdot e^{inx} \\ &= \sum_{n=-\infty}^{\infty} F[n]\cdot e^{inx} \quad \mbox{(engineering)} \end{align}

In various fields of science, the sequence has other names, such as characteristic function (probability theory). In engineering, particularly when variable x represents time, the 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 [a,b]

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

Failed to parse (Missing texvc executable; please see math/README to configure.): g(x)=\sum_{n=-\infty}^\infty G[n]\cdot e^{i 2\pi \frac{n}{\tau} x}\,

is a periodic function, whose period is Failed to parse (Missing texvc executable; please see math/README to configure.): \tau\,

on the domain Failed to parse (Missing texvc executable; please see math/README to configure.): \{x \isin \mathbb{R}\}

. If a function is square-integrable in the interval Failed to parse (Missing texvc executable; please see math/README to configure.): [a,\ a+\tau],   it can be represented in that interval by the formula above, and the Fourier coefficients are given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \tau\, -periodic, then Failed to parse (Missing texvc executable; please see math/README to configure.): a\,

is an arbitrary choice.  Two popular choices are Failed to parse (Missing texvc executable; please see math/README to configure.): a=0\,
and Failed to parse (Missing texvc executable; please see math/README to configure.): a=-\tau/2.\,

  Also note that the Fourier transform, which can be expressed^[2] in terms of the Fourier series coefficients, is an alternative and commonly used frequency domain representation.

Fourier series on a square

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

Failed to parse (Missing texvc executable; please see math/README to configure.): f(x,y) = \sum_{j,k \in \mathbb{Z}} c_{j,k}e^{ijx}e^{iky},

Failed to parse (Missing texvc executable; please see math/README to configure.): c_{j,k} = {1 \over 4 \pi^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} f(x,y) e^{-ijx}e^{-iky}\, dx \, dy\ .

Aside from being useful for solving partial differential equations such as the heat equation, one notable application of Fourier series of the square is in image compression. 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

Main article: Hilbert space

In the language of Hilbert spaces, the set of functions Failed to parse (Missing texvc executable; please see math/README to configure.): \{ e_n = e^{i n x},n\in\mathbb{Z}\}

is an orthonormal basis for the space Failed to parse (Missing texvc executable; please see math/README to configure.): L^2([-\pi,\pi])
of square-integrable functions of Failed to parse (Missing texvc executable; please see math/README to configure.): [-\pi,\pi]

. This space is actually a Hilbert space with an inner product given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \langle f, g \rangle \ \stackrel{\mathrm{def}}{=} \ \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\overline{g(x)}\,dx.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): 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. Clearly, the sines and cosines form an orthonormal set:

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\pi}^{\pi} \cos(mx)\, \cos(nx)\, dx = \pi \delta_{mn},

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\pi}^{\pi} \sin(mx)\, \sin(nx)\, dx = \pi \delta_{mn}

(where Failed to parse (Missing texvc executable; please see math/README to configure.): \delta_{mn}

is the Kronecker delta), and

Failed to parse (Missing texvc executable; please see math/README to configure.): \int_{-\pi}^{\pi} \cos(mx)\, \sin(nx)\, dx = 0.

The density of their span is a consequence of the Stone-Weierstrass theorem.

Properties

We say that Failed to parse (Missing texvc executable; please see math/README to configure.): f \in C^k(\mathbb{T})

if Failed to parse (Missing texvc executable; please see math/README to configure.): f
is a function of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{R}
which is Failed to parse (Missing texvc executable; please see math/README to configure.): k
times differentiable, its kth derivative is continuous, and is Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi

-periodic.

• If f is a Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi

-periodic odd function, then Failed to parse (Missing texvc executable; please see math/README to configure.): a_n=0

for all Failed to parse (Missing texvc executable; please see math/README to configure.): n

.

• If f is a Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi

-periodic even function, then Failed to parse (Missing texvc executable; please see math/README to configure.): b_n=0

for all Failed to parse (Missing texvc executable; please see math/README to configure.): n

.

• If f is square-integrable, Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n\rightarrow +\infty}c_n=0

, Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n\rightarrow +\infty}c_{-n}=0 , Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n\rightarrow +\infty}a_n=0

and Failed to parse (Missing texvc executable; please see math/README to configure.): \lim_{n\rightarrow +\infty}b_n=0.

• If Failed to parse (Missing texvc executable; please see math/README to configure.): f \in C^1(\mathbb{T})

, then the Fourier coefficients Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{f'}(n)

of the derivative Failed to parse (Missing texvc executable; please see math/README to configure.): f'(t)
can be expressed in terms of the Fourier coefficients Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{f}(n)
of the function Failed to parse (Missing texvc executable; please see math/README to configure.): f(t)

, via the formula Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{f'}(n) = in \hat{f}(n) .

• If Failed to parse (Missing texvc executable; please see math/README to configure.): f \in C^k(\mathbb{T})

, then Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{f^{(k)}}(n) = (in)^k \hat{f}(n) . In particular, since Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{f^{(k)}}(n)

tends to zero, we have that Failed to parse (Missing texvc executable; please see math/README to configure.): |n|^k\hat{f}(n)
tends to zero, which means that the Fourier coefficients converge to zero faster than the kth power of n.

• THEOREM (Parseval). If Failed to parse (Missing texvc executable; please see math/README to configure.): f \in L^2([-\pi,\pi])

, then Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{n=-\infty}^{\infty} |\hat{f}(n)|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi} |f(x)|^2 \, dx .

• THEOREM (Plancherel). If Failed to parse (Missing texvc executable; please see math/README to configure.): c_0,\, c_{\pm 1},\, c_{\pm 2},\ldots

are coefficients and Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{n=-\infty}^\infty |c_n|^2 < \infty
then there is a unique function Failed to parse (Missing texvc executable; please see math/README to configure.): f\in L^2([-\pi,\pi])
such that Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{f}(n) = c_n
for every Failed to parse (Missing texvc executable; please see math/README to configure.): n

.

• The Convolution theorem states that if f and g are in Failed to parse (Missing texvc executable; please see math/README to configure.): L^2([-\pi,\pi])

, then Failed to parse (Missing texvc executable; please see math/README to configure.): \widehat{f*g}(n) = \hat{f}(n)\hat{g}(n) , where Failed to parse (Missing texvc executable; please see math/README to configure.): f*g

denotes the Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi

-periodic convolution of f and g.

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

Main articles: Generalized function and Distribution (mathematics)

One can extend the notion of Fourier coefficients to functions which are not square-integrable, and even to objects which are not functions. This is very useful in engineering and applications because we often need to take the Fourier transform of a Dirac delta function. The Dirac delta Failed to parse (Missing texvc executable; please see math/README to configure.): \delta

is not actually a function, it is a measure but it still has a Fourier transform, and Failed to parse (Missing texvc executable; please see math/README to configure.): \hat{\delta}(n)={1 \over 2\pi}
for every Failed to parse (Missing texvc executable; please see math/README to configure.): n

. This generalization enlarges the domain of definition of the Fourier transform from Failed to parse (Missing texvc executable; please see math/README to configure.): L^2([-\pi,\pi])

to a superset of Failed to parse (Missing texvc executable; please see math/README to configure.): L^2

. The Fourier series converges weakly.

Compact groups

Main articles: Compact group and Lie group

One of the interesting properties of the Fourier transform which we have mentioned, is that it carries convolutions to pointwise products. 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 Failed to parse (Missing texvc executable; please see math/README to configure.): L^2(G) , where Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): [-\pi,\pi]
case.

Riemannian manifolds

Main articles: Laplace operator and Riemannian manifold

If the domain is not a group, then there is no intrinsically defined convolution. However, if Failed to parse (Missing texvc executable; please see math/README to configure.): X

is a compact Riemannian manifold, it has a Laplace-Beltrami operator. Since the Laplace-Beltrami operator is the differential operator that corresponds to the heat equation for the Riemannian manifold Failed to parse (Missing texvc executable; please see math/README to configure.): X

, and 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 Failed to parse (Missing texvc executable; please see math/README to configure.): L^2(X) , where Failed to parse (Missing texvc executable; please see math/README to configure.): X

is a Riemannian manifold. The Fourier series converges in ways similar to the Failed to parse (Missing texvc executable; please see math/README to configure.): [-\pi,\pi]
case. A typical example is X=the sphere, in which case the Fourier basis consists of spherical harmonics.

Locally compact Abelian groups

Main article: Pontryagin duality

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 Failed to parse (Missing texvc executable; please see math/README to configure.): L^1(G)

or Failed to parse (Missing texvc executable; please see math/README to configure.): L^2(G)

, where G is an LCA group. If Failed to parse (Missing texvc executable; please see math/README to configure.): G

is compact, one also obtains a Fourier series, which converges similarly to the Failed to parse (Missing texvc executable; please see math/README to configure.): [-\pi,\pi]
case, but if Failed to parse (Missing texvc executable; please see math/README to configure.): G
is noncompact, one obtains instead a Fourier integral. This generalization yields the usual Fourier transform when the underlying locally compact Abelian group is Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{-\infty}^\infty

by a finite one,

Failed to parse (Missing texvc executable; please see math/README to configure.): S_N(x) = \sum_{n=-N}^N \hat{f}(n) e^{inx}

.

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

Least squares property

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

Failed to parse (Missing texvc executable; please see math/README to configure.): p(x)=\sum_{n=-N}^N p_n e^{inx}.

Note that Failed to parse (Missing texvc executable; please see math/README to configure.): S_N(x)

is a trigonometric polynomial of degree N. Parseval's theorem implies that

THEOREM. Failed to parse (Missing texvc executable; please see math/README to configure.): S_N(x)

is the unique best trigonometric polynomial of degree Failed to parse (Missing texvc executable; please see math/README to configure.): N
approximating Failed to parse (Missing texvc executable; please see math/README to configure.): f(x)

, in the sense that, for any trigonometric polynomial Failed to parse (Missing texvc executable; please see math/README to configure.): p\neq S_N

of degree Failed to parse (Missing texvc executable; please see math/README to configure.): N

, we have Failed to parse (Missing texvc executable; please see math/README to configure.): \|S_N - f\| \lneqq \|p - f\| .

Here, the Hilbert space norm is

Failed to parse (Missing texvc executable; please see math/README to configure.): \| g \| = \sqrt{{1 \over 2\pi} \int_{-\pi}^{\pi} |g(x)|^2 \, dx}.

Convergence

Main article: Convergence of Fourier series

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

THEOREM. If Failed to parse (Missing texvc executable; please see math/README to configure.): f\in L^2([-\pi,\pi]) , then the Fourier series converges in Failed to parse (Missing texvc executable; please see math/README to configure.): L^2([-\pi,\pi]) , i.e., Failed to parse (Missing texvc executable; please see math/README to configure.): \|S_N - f\|

converges to 0 as Failed to parse (Missing texvc executable; please see math/README to configure.): N
goes to infinity.

We have already mentioned that if f is twice continuously differentiable, then Failed to parse (Missing texvc executable; please see math/README to configure.): n^2 \hat{f}(n)

converges to zero as n goes to infinity. This immediately gives a second convergence result.

THEOREM. If Failed to parse (Missing texvc executable; please see math/README to configure.): f \in C^2(\mathbb{T}) , then Failed to parse (Missing texvc executable; please see math/README to configure.): \sup_x |f(x) - S_N(x)| \leq \sum_{|n|>N} |\hat{f}(n)|

converges to zero, i.e., Failed to parse (Missing texvc executable; please see math/README to configure.): S_N
converges to Failed to parse (Missing texvc executable; please see math/README to configure.): f
uniformly.

In particular, Failed to parse (Missing texvc executable; please see math/README to configure.): S_N

converges to f pointwise.

Many further cases are discussed in the main article, Convergence of Fourier series, ranging from the moderately simple result that the series converges at x if f is differentiable at x, to Lennart Carleson's much more sophisticated result that the Fourier series of an Failed to parse (Missing texvc executable; please see math/README to configure.): L^2

function actually converges almost everywhere.

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. This function is not in Failed to parse (Missing texvc executable; please see math/README to configure.): L^2([-\pi,\pi]) .

See also

• Fourier transform
• Harmonic analysis
• Gibbs phenomenon
• Sturm-Liouville theory
• Laurent series — the substitution Failed to parse (Missing texvc executable; please see math/README to configure.): q=e^{i x}

transforms a Fourier series into a Laurent series, or conversely. This is used in the q-series expansion of the j-invariant.

Notes

1. ^ http://gallica.bnf.fr/scripts/ConsultationTout.exe?O=03370&E=00000220&N=7
2. ^ The Fourier transform is given by:

   Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} G(f) &= \int_{-\infty}^{\infty} g(x)\cdot e^{-i2\pi f x} dx \\ &= \int_{-\infty}^{\infty} \left[\sum_{n=-\infty}^{\infty} G[n]\cdot e^{i 2\pi \frac{n}{\tau} x}\right]\cdot e^{-i2\pi f x} dx \\ &= \sum_{n=-\infty}^{\infty} G[n]\cdot \int_{-\infty}^\infty e^{-i2\pi(f-\frac{n}{\tau})x} dx \\ &= \sum_{n=-\infty}^{\infty} G[n]\cdot \delta \left(f-\frac{n}{\tau}\right) \end{align}

   where variable Failed to parse (Missing texvc executable; please see math/README to configure.): f\, represents a continuous frequency domain. This formula is a Dirac comb modulated by the Fourier series coefficients.

References

• Joseph Fourier, translated by Alexander Freeman (published 1822, translated 1878, re-released 2003). The Analytical Theory of Heat. Dover Publications. ISBN 0-486-49531-0.  2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's work Théorie Analytique de la Chaleur, originally published in 1822.
• Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition. Dover Publications, Inc., New York, 1976. ISBN 0-486-63331-4
• Felix Klein, Development of mathematics in the 19th century. Mathsci Press Brookline, Mass, 1979. Translated by M. Ackerman from Vorlesungen über die Entwicklung der Mathematik im 19 Jahrhundert, Springer, Berlin, 1928.
• Walter Rudin, Principles of mathematical analysis, Third edition. McGraw-Hill, Inc., New York, 1976. ISBN 0-07-054235-X
• William E. Boyce and Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems, Eighth edition. John Wiley & Sons, Inc., New Jersey, 2005. ISBN 0-471-43338-1

External links

• Phasor Phactory Allows custom control of the harmonic amplitudes for arbitrary terms
• Java applet shows Fourier series expansion of an arbitrary function
• Example problems - Examples of computing Fourier Series
• Fourier series explanation - A simple, non-mathematical approach
• Eric W. Weisstein, Fourier Series at MathWorld.
• Fourier Series Module by John H. Mathews
• Joseph Fourier - A site on Fourier's life which was used for the historical section of this article

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