Fourier Series

The Fourier series is a Mathematical tool used for analyzing Periodic Function s by decomposing such a function into a weighted sum of much simpler Sinusoidal component functions sometimes referred to as '''normal Fourier modes''', or simply '''modes''' for short. The weights, or coefficients, of the modes, are a One-to-one mapping of the original function. Generalizations include Generalized Fourier Series and other expansions over Orthonormal Bases .

Fourier series serve many useful purposes, as manipulation and conceptualization of the modal coefficients are often easier than with the original function. Areas of application include Electrical Engineering , Vibration analysis, Acoustics , Optics , Signal and Image Processing , and Data Compression . Using the tools and techniques of Spectroscopy , for example, astronomers can deduce the chemical composition of a star by analyzing the frequency components, or Spectrum , of the star's emitted light. Similarly, engineers can optimize the design of a telecommunications system using information about the spectral components of the data signal that the system will carry (see also Spectrum Analyzer ).

The Fourier series is named after the French scientist and mathematician Joseph Fourier , who used them in his influential work on Heat Conduction , ''Théorie Analytique de la Chaleur'' (''The Analytical Theory of Heat''), published in 1822.

DEFINITION

General form

Given a , Periodic with period ''T'', and Square-integrable over the interval from

t_1

t_2

of length ''T'', that is,

Notice that ''a_n'' are 0 because the

x\mapsto x\cos(nx)

are even functions. Hence the Fourier series for this function is:

:

f(x)=rac{a_0}{2} + \sum_{n=1}^{\infty}\left[a_n\cos\left(nx
ight)+b_n\sin\left(nx
ight)
ight]

=2\sum_{n=1}^{\infty}rac{(-1)^{n+1}}{n} \sin(nx), \quad orall x\in

{Link without Title} .

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:
:

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

which yields:

\sum_{n>0}rac{1}{n^2}=rac{\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( rac{n\pi}{L} x 
ight)

where

:

b_n =  rac{2}{L} \int_0^L f(x) \sin \left( rac{n\pi}{L} x
ight)\, 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( rac{(2n +1)\pi}{2L} x 
ight)

where

:

b_n =  rac{2}{L} \int_0^L f(x) \sin \left( rac{(2n+1)\pi}{2L} x
ight)\, dx.

Interpretation: decomposing a movement in rotations

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}
ight]

to the point

B=\left[\begin{matrix}u(t+dt)\v(t+dt)\end{matrix}
ight]

, 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}
ight]

. As a result, we can write ''f'' as:
:

f(t)=\left[\begin{matrix}u(dt)-u(0)\v(dt)-v(0)\end{matrix}
ight]+\left[\begin{matrix}u(2dt)-u(dt)\v(2dt)-v(dt)\end{matrix}
ight]+\cdots+\left[\begin{matrix}u(t+dt)-u(t)\v(t+dt)-v(t)\end{matrix}
ight]

:::

=\int_0^trac{1}{dx}\left[\begin{matrix}u(x+dx)-u(x)\v(x+dx)-v(x)\end{matrix}
ight]\,dx.

Now instead of seeing ''f'' as a sum of infinitesimal translations, we can see it as an infinite sum of rotations of different radii. This interpretation is convenient, in particular when the movement is periodic.

Let

\chi_n=e^{inx}

be the ''n''-turn per second rotation (of radius 1) (sometimes called Character ). We want to write ''f'' as

f(x)=\sum c_n \chi_n

. We can prove (see mathematical derivation below) that the radii of the rotations (the coefficients

c_n

) are exactly the ones we gave in the previous paragraph.

For example, the plot of the function

f:t\mapsto 2\cos\left(rac{t}{2}
ight)e^{rac{3}{2}it}

is closed, which means the function is periodic. The loop in the curve suggests that it is the sum of two periodic functions, one having a shorter period than the other. Indeed, it can be written:

f(t)=e^{it}+e^{2it}=\chi_1(t)+\chi_2(t)

. All its Fourier coefficients are zero except

c_1=1

and

c_2=1

. The graphical interpretation of a rotation is much harder to do than that of the translations because instead of visually seeing the movement from one point to another we have to add the whole motion for the decomposition to make sense (we are reasoning in rotation frequencies rather than in time).

Mathematically, adopting this point of view is seeing Fourier series as a tool to understand Linear Operators that commute with translations. The functions

\chi_n

are precisely the multiplicative characters of the group

\mathbb{R}/2\pi\mathbb{Z}

.

HISTORICAL DEVELOPMENT

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 :

:: $arphi(y)=a\cosrac{\pi y}{2}+a'\cos 3rac{\pi y}{2}+a''\cos5rac{\pi y}{2}+\cdots.$

:Multiplying both sides by $\cos(2i+1)rac{\pi y}{2}$ , and then integrating from $y=-1$ to $y=+1$ yields:

:: $a_i=\int_{-1}^1arphi(y)\cos(2i+1)rac{\pi y}{2}\,dy.$

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'':R → '''C''' is a function, we would like to write this function as a sum of trigonometric functions, i.e.

f(x)=\sum c_n e^{inx}

. We have to restrict our choice of functions in order for this to make sense. First of all, if ''f'' has period ''T'', then by changing variables, one can study

x\mapsto f\left(rac{T}{2\pi}x
ight)

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(rac{T}{2\pi}x
ight)

to any interval of length 2π, {Link without Title} , say.

Where <math>\overline{f(x)}</math> Denotes The "http://wwwinformationdelightinfo/information/entry/Complex_conjugate" class="copylinks">Conjugate of ''f''(''x'') We will denote by <math>\ \cdot \</math> the associated Norm

We usually define

orall n\in\mathbb{Z}, c_n=\left\langle f,e^{i n x}
ight
angle

. These numbers are called complex Fourier coefficients. Their expression is

:

c_n = rac{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 ''c''_''n'' coefficient. This reminds one of the formulae

:

\cos x =rac{e^{ix}+e^{-ix}}{2}{
m~~~~and~~~~}\sin x =rac{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 n x}+c_n e^{i n x}
ight .

After replacing ''c''_n by its expression and simplifying the result we get

:

f(x)=c_0+\sum_{n>0}\left[rac{1}{\pi}\left(\int_{-\pi}^\pi f(t)\cos\left(n t
ight)\, dt
ight)\cos\left(n x
ight)+rac{1}{\pi}\left(\int_{-\pi}^\pi f(t)\sin\left(n t
ight)\, dt
ight)\sin\left(n x
ight)
ight].

If, for a non-negative integer ''n'', we define the real Fourier coefficients ''a''_''n'' and ''b''_''n'' by

:

a_n = rac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos\left(n x
ight)\, dx,

b_n = rac{1}{\pi}\int_{-\pi}^{\pi} f(x) \sin\left(n x
ight)\, dx,

we get:

:

f(x)=rac{a_0}{2}+\sum_{n>0}\left x
ight)+b_n\sin\left(n x
ight)
ight .

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=rac{a_n-ib_n}{2} \mbox{ and }c_{-n}=rac{a_n+ib_n}{2}\mbox{ for all } n.

If ''f'' is an Odd Function , then $a_n=0$ for all $n$ because $f(x)\cos\left(n\pirac{x}{T}
ight)$ is then also odd, so its integral on $T$ is zero. If ''f'' is an Even Function , then $b_n=0$ for a similar reason.

If ''f'' is piecewise continuous, $\lim_{n
ightarrow +\infty}c_n(f)=0$ , $\lim_{n
ightarrow +\infty}c_{-n}(f)=0$ , $\lim_{n
ightarrow +\infty}a_n(f)=0$ and $\lim_{n
ightarrow +\infty}b_n(f)=0.$

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)}
ight)=(in)^k c_n(f),

where

f^{(k)}

denotes the ''k''th derivative of ''f''.

For any positive integer ''k'', if ''f'' is C^{''k'' − 1} and piecewise C^''k'', then

For example, the Fourier of this article are obtained by taking ''G'' = R/2π'''Z'''. 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 = rac{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 T^''n'' = '''R'''^''n''/(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'' = T^''n'' gives the corresponding Fourier series.

APPROXIMATION AND CONVERGENCE OF FOURIER SERIES

Definition of a Fourier series

Let

\chi_n(x)=e^{in\pi rac{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

We Have <math>\f-p\^2 \f\^2-2\mbox{Re}\langle f,p angle+\p\^2</math>, where Re(''z'') denotes the real part of ''z''
:<math>\p\^2 \sum_{n=-N}^Nx_n^2</math>
:<math>\f-p\^2 \f\^2+\sum_{n=-N}^N\left[c_n-x_n^2-c_n^2 ight]</math>
:<math>\f-f N\ \min_{p\in\mathcal{T}_N}\left\{\f-p\,p\in\mathcal{T}_N ight\},</math>

Convergence

See Also: Convergence of Fourier series

While the Fourier coefficients ''a''_''n'' and ''b''_''n'' 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

C N\,\chi N(x) Ight^2\,dx 0</math>

In 1922, Andrey Kolmogorov published an article entitled Une série de Fourier-Lebesgue divergente presque partout in which he gave an example of a Lebesgue-integrable function whose Fourier series diverges almost everywhere. This function is not in

L^2(\mu)

.

Plancherel's and Parseval's theorems

Another important property of the Fourier series is the Plancherel Theorem . Let

f,g\in L^2(\mu)

and

c_n(f), c_n(g)

be the corresponding complex Fourier coefficients. Then

:

\sum_{n\in\mathbb{Z}} c_n(f)\overline{c_n(g)} = rac{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:

:<math>Rac{a 0^2}{4} + Rac{1}{2} \sum {n 1}^\infty \left( a_n^2 + b_n^2 ight) = rac{1}{2T} \int_{-T}^T f(x)^2\, dx</math>

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Information About

Fourier Series