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Fourier series |
The
first four Fourier series approximations for a square wave.
In mathematics,
a Fourier series decomposes periodic
functions or periodic
signals into the sum of a (possibly infinite) set of simple oscillating
functions, namely sines and
cosines (or complex exponentials). The study of
Fourier series is a branch of Fourier analysis.
The 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.
Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his
initial results in his 1807 Mémoire sur la
propagation de la chaleur dans les corps solides (Treatise on the propagation of
heat in solid bodies), and publishing his Théorie
analytique de la chaleur in
1822. Early ideas of decomposing a periodic function into the sum of simple
oscillating functions date back to the 3rd century BC, when ancient astronomers
proposed an empiric model of planetary motions, based on deferents and epicycles.
The heat equation is a partial differential equation. Prior to
Fourier's work, no solution to the heat equation was known in the general case,
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 (orlinear 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.
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.
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, and especially
those involving linear differential equations with constant coefficients, for
which the eigensolutions are sinusoids. The Fourier
series has many such applications in electrical engineering, vibration analysis, acoustics,optics, signal
processing, image processing, quantum
mechanics, econometrics,[1] thin-walled shell theory,[2] etc.
“ |
Multiplying both sides by |
” |
—Joseph Fourier, Mémoire sur la
propagation de la chaleur dans les corps solides. (1807)[3][nb 1] |
This immediately gives any coefficient of the trigonometrical series for
for any function which has such an
expansion. It works because if
has such an expansion, then (under
suitable convergence assumptions) the integral
can be carried out term-by-term. But
all terms involving for j ≠ k vanish
when integrated from −1 to 1, leaving only the kth term.
In these few lines, which are close to the
modern formalism used in Fourier series, Fourier
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 any arbitrary function. 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 a later
competition essay in 1811, 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 evenrigour.
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 thebasis 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, or for functions on a bounded (compact) interval.
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 [−π, π],
as Fourier did (see the quote above), and we will then discuss different
formulations and generalizations.
For a periodic function ƒ(x) that is integrable
on [−π, π], the numbers
and
are called the Fourier coefficients of ƒ. One introduces the partial sums of the Fourier series for ƒ,
often denoted by
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
is called the Fourier series of ƒ.
These trigonometric functions can themselves be expanded, using multiple angle formulae.
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 [−π, π],
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.[4] 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.
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
In this case, the Fourier coefficients
are given by
It can be proven that the Fourier
series converges to ƒ(x)
at every point x where ƒ is differentiable, and therefore:
|
|
(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 in example 1 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 (x, y)
∈
[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(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
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(x, y) is
nontrivial. The function T cannot be written as a closed-form expression. This method of
solving the heat problem was 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
integern.
We can use Euler's formula,
where i is the imaginary unit,
to give a more concise formula:
Assuming f(x) is a periodic function
with T = 2π, the
Fourier coefficients are then given by:
The Fourier coefficients an, bn, cn are related via
and
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 or F,
and functional notation often replaces subscripting:
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.
The following formula, with appropriate
complex-valued coefficients G[n],
is a periodic function with period τ on all of R:
If a function is square-integrable in the interval [a, a + τ],
it can be represented in that interval by the formula above. I.e., when
the coefficients are derived from a function, h(x),
as follows:
then g(x)
will equal h(x) in
the interval [a,a+τ ].
It follows that if h(x)
is τ-periodic, then:
§
g(x)
and h(x) are equal
everywhere, except possibly at discontinuities, and
§
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:
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. g(x)
can be recovered from this representation by an inverse Fourier transform:
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 at the harmonic frequencies.[nb 2]
We can also define the Fourier series
for functions of two variables x and y in the square [−π, π]×[−π, π]:
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.
Main article: Hilbert space
In the language of Hilbert spaces,
the set of functions is an orthonormal
basis for the space
of
square-integrable functions of
.
This space is actually a Hilbert space with an inner product given for any two elements f and g by:
The basic Fourier series result for
Hilbert spaces can be written as
Sines and cosines form an orthonormal set, as illustrated
above. The integral of sine, cosine and their product is zero (green and red
areas are equal, and cancel out) when m and n are different, and pi if they are
equal.
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 anorthogonal set:
(where is the Kronecker delta),
and
furthermore, the sines and cosines are
orthogonal to the constant function 1. An orthonormal basis forL2([−π, π])
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.
We say that ƒ belongs to 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
for all n.
§
If ƒ is a 2π-periodic even function,
then
for all n.
§
If ƒ is integrable, ,
and
This result is known as the Riemann–Lebesgue lemma.
§
A doubly infinite sequence in
is
the sequence of Fourier coefficients of a function in
if and only if it is a convolution of
two sequences in
. See [1]
§
If , then the Fourier coefficients
of
the derivative
can be expressed in terms of the
Fourier coefficients
of the function
,
via the formula
.
§
If , then
.
In particular, since
tends
to zero, we have that
tends
to zero, which means that the Fourier coefficients converge to zero faster than
the kth power of n.
§
Parseval's theorem. If , then
.
§
Plancherel's theorem. If are coefficients and
then there is a unique function
such
that
for
every n.
§
The first convolution theorem states that if ƒ and g are in L1([−π, π]),
then ,
where ƒ ∗ g denotes the 2π-periodic convolution of ƒ and g.
(The factor
is not necessary for 1-periodic
functions.)
§
The second convolution theorem states that .
§
The Poisson summation formula states that the periodic summation of a function,
has a Fourier series representation whose coefficients are proportional to
discrete samples of the continuous
Fourier transform of
:
.
Similarly, the periodic summation of has a Fourier series representation
whose coefficients are proportional to discrete samples of
,
a fact which provides a pictorial understanding of aliasing and the
famous sampling theorem.
§
Also see Fourier_analysis#Variants_of_Fourier_analysis.
Main articles: Compact group, Lie group,
and Peter–Weyl theorem
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 [−π, π]
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.
The atomic orbitals of chemistry arespherical harmonics and can be used to produce Fourier
series on the sphere.
Main articles: Laplace operator and Riemannian manifold
If the domain is not a group, then
there is no intrinsically defined convolution. However, if X is a compact Riemannian manifold, it has aLaplace–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 [−π, π]
case. A typical example is to take X to be the sphere with the usual
metric, in which case the Fourier basis consists ofspherical harmonics.
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 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 [−π, π]
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 .
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 by a finite one,
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.
We say that p is a trigonometric polynomial of degree N when it is of the form
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 of degree N, we have
Here, the Hilbert space norm is
Main article: Convergence of Fourier series
See also: Gibbs phenomenon
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, converges
to 0 as N goes to infinity.
We have already mentioned that if ƒ is continuously differentiable,
then 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 , 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 tends
to zero as
.
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
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, toLennart 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 specify the convergence conditions, are sometimes referred
to generically as "Fourier's theorem" or "the Fourier
theorem".[5][6][7][8]
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. The uniform boundedness principle yields a simple non-constructive proof
of this fact.
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).
§
Laurent series — the substitution q = eix transforms a Fourier series into a
Laurent series, or conversely. This is used in the q-series expansion of the j-invariant.