Fourier series

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In mathematics, a Fourier series ( [1] ) is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle (or period) of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.

History

The Fourier series is named in honor of Jean-Baptiste 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. [upper-alpha 1] 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 (Analytical theory of heat) in 1822. The Mémoire introduced Fourier analysis, specifically Fourier series. Through Fourier's research the fact was established that an arbitrary (at first, continuous [2] and later generalized to any piecewise-smooth [3] ) function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier in 1807, before the French Academy. [4] 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 (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.

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, Peter Gustav Lejeune Dirichlet [5] and Bernhard Riemann [6] [7] [8] 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, [9] shell theory, [10] etc.

Definition

Consider a real-valued function, ${\displaystyle s(x),}$ that is integrable on an interval of length ${\displaystyle P,}$ which will be the period of the Fourier series. The correlation function:

${\displaystyle \mathrm {X} _{f}(\tau )\triangleq \int _{P}s(x)\cdot \cos \left(2\pi f(x-\tau )\right)\,dx;\quad \tau \in \left[\ 0,\ 2\pi /f\ \right]}$

is essentially a matched filter, with template${\displaystyle \cos(2\pi fx).}$ Its peak value is a relative measure of the presence of frequency ${\displaystyle f}$ in function ${\displaystyle s.}$ The analysis process determines, for certain key frequencies, the maximum correlation and the corresponding phase offset, ${\displaystyle (\tau f).}$ The synthesis process (the actual Fourier series), in terms of parameters to be determined by analysis, is:

Fourier series, amplitude-phase form
${\displaystyle s_{\scriptscriptstyle N}(x)={\frac {A_{0}}{2}}+\sum _{n=1}^{N}A_{n}\cdot \cos \left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right).}$

(Eq.1)

• In general, integer ${\displaystyle N}$ is theoretically infinite. Even so, the series might not converge or exactly equate to ${\displaystyle s(x)}$ at all values of ${\displaystyle x}$ (such as a single-point discontinuity) in the analysis interval. For the "well-behaved" functions typical of physical processes, equality is customarily assumed.
• Integer ${\displaystyle n,}$ used as an index, is also the number of cycles of the ${\displaystyle n}$-th harmonic in interval ${\displaystyle P.}$ Therefore, the length of a cycle, in the units of ${\displaystyle x,}$ is ${\displaystyle P/n.}$ The corresponding harmonic frequency is ${\displaystyle n/P,}$ so the ${\displaystyle n}$-th harmonic is ${\displaystyle \cos \left(2\pi x{\tfrac {n}{P}}\right).}$ Some texts define ${\displaystyle P=2\pi }$ to simplify the argument of the sinusoid functions at the expense of generality.

Rather than computationally intensive cross-correlation, Fourier analysis customarily exploits a trigonometric identity:

${\displaystyle A_{n}\cdot \cos \left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right)\ \equiv \ \underbrace {A_{n}\cos(\varphi _{n})} _{a_{n}}\cdot \cos \left({\tfrac {2\pi }{P}}nx\right)+\underbrace {A_{n}\sin(\varphi _{n})} _{b_{n}}\cdot \sin \left({\tfrac {2\pi }{P}}nx\right),}$

where parameters ${\displaystyle a_{n}}$ and ${\displaystyle b_{n}}$ replace ${\displaystyle A_{n}}$ and ${\displaystyle \varphi _{n},}$ and can be found by evaluating the cross-correlation at only two values of phase: [11]

Fourier coefficients
{\displaystyle {\begin{aligned}a_{n}&={\frac {2}{P}}\int _{P}s(x)\cdot \cos \left({\tfrac {2\pi }{P}}nx\right)\,dx\\b_{n}&={\frac {2}{P}}\int _{P}s(x)\cdot \sin \left({\tfrac {2\pi }{P}}nx\right)\,dx.\end{aligned}}}

(Eq.2)

Then:${\displaystyle A_{n}={\sqrt {a_{n}^{2}+b_{n}^{2}}}}$ and ${\displaystyle \varphi _{n}=\operatorname {arctan2} (b_{n},a_{n})}$ (see Atan2) or more directly:

Fourier series, sine-cosine form
${\displaystyle s_{\scriptscriptstyle N}(x)={\frac {a_{0}}{2}}+\sum _{n=1}^{N}\left(a_{n}\cos \left({\tfrac {2\pi }{P}}nx\right)+b_{n}\sin \left({\tfrac {2\pi }{P}}nx\right)\right).}$

(Eq.3)

And note that ${\displaystyle a_{0}}$ and ${\displaystyle b_{0}}$ can be reduced to  ${\displaystyle a_{0}={\frac {2}{P}}\int _{P}s(x)\,dx}$  and  ${\displaystyle b_{0}=0.}$

Another applicable identity is Euler's formula. Here, complex conjugation is denoted by an asterisk:

${\displaystyle {\begin{array}{lll}\cos \left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right)&{}\equiv {\tfrac {1}{2}}e^{i\left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right)}&{}+{\tfrac {1}{2}}e^{-i\left({\tfrac {2\pi }{P}}nx-\varphi _{n}\right)}\\&=\left({\tfrac {1}{2}}e^{-i\varphi _{n}}\right)\cdot e^{i{\tfrac {2\pi }{P}}(+n)x}&{}+\left({\tfrac {1}{2}}e^{-i\varphi _{n}}\right)^{*}\cdot e^{i{\tfrac {2\pi }{P}}(-n)x}.\end{array}}}$

Therefore, with definitions:

${\displaystyle c_{n}\triangleq \left\{{\begin{array}{lll}A_{0}/2&=a_{0}/2,\quad &n=0\\{\tfrac {A_{n}}{2}}e^{-i\varphi _{n}}&={\tfrac {1}{2}}(a_{n}-ib_{n}),\quad &n>0\\c_{|n|}^{*},\quad &&n<0\end{array}}\right\}\quad =\quad {\frac {1}{P}}\int _{P}s(x)\cdot e^{-i{\tfrac {2\pi }{P}}nx}\ dx,}$

the final result is:

Fourier series, exponential form
${\displaystyle s_{_{N}}(x)=\sum _{n=-N}^{N}c_{n}\cdot e^{i{\tfrac {2\pi }{P}}nx}.}$

(Eq.4)

This is the customary form for generalizing to complex-valued ${\displaystyle s(x)}$ (next section).

Complex-valued functions

If ${\displaystyle s(x)}$ is a complex-valued function of a real variable ${\displaystyle x,}$ both components (real and imaginary part) are real-valued functions that can be represented by a Fourier series. The two sets of coefficients and the partial sum are given by:

${\displaystyle c_{_{Rn}}={\frac {1}{P}}\int _{P}\operatorname {Re} \{s(x)\}\cdot e^{-i{\tfrac {2\pi }{P}}nx}\ dx}$   and   ${\displaystyle c_{_{In}}={\frac {1}{P}}\int _{P}\operatorname {Im} \{s(x)\}\cdot e^{-i{\tfrac {2\pi }{P}}nx}\ dx}$
${\displaystyle s_{_{N}}(x)=\sum _{n=-N}^{N}c_{_{Rn}}\cdot e^{i{\tfrac {2\pi }{P}}nx}+i\cdot \sum _{n=-N}^{N}c_{_{In}}\cdot e^{i{\tfrac {2\pi }{P}}nx}=\sum _{n=-N}^{N}\left(c_{_{Rn}}+i\cdot c_{_{In}}\right)\cdot e^{i{\tfrac {2\pi }{P}}nx}.}$

Defining ${\displaystyle c_{n}\triangleq c_{_{Rn}}+i\cdot c_{_{In}}}$ yields:

${\displaystyle s_{_{N}}(x)=\sum _{n=-N}^{N}c_{n}\cdot e^{i{\tfrac {2\pi }{P}}nx}.}$

(Eq.5)

This is identical to Eq.4 except ${\displaystyle c_{n}}$ and ${\displaystyle c_{-n}}$ are no longer complex conjugates. The formula for ${\displaystyle c_{n}}$ is also unchanged:

{\displaystyle {\begin{aligned}c_{n}&={\frac {1}{P}}\int _{P}\operatorname {Re} \{s(x)\}\cdot e^{-i{\tfrac {2\pi }{P}}nx}\ dx+i\cdot {\frac {1}{P}}\int _{P}\operatorname {Im} \{s(x)\}\cdot e^{-i{\tfrac {2\pi }{P}}nx}\ dx\\[4pt]&={\frac {1}{P}}\int _{P}\left(\operatorname {Re} \{s(x)\}+i\cdot \operatorname {Im} \{s(x)\}\right)\cdot e^{-i{\tfrac {2\pi }{P}}nx}\ dx\ =\ {\frac {1}{P}}\int _{P}s(x)\cdot e^{-i{\tfrac {2\pi }{P}}nx}\ dx.\end{aligned}}}

Other common notations

The notation ${\displaystyle c_{n}}$ is inadequate for discussing the Fourier coefficients of several different functions. Therefore, it is customarily replaced by a modified form of the function (${\displaystyle s}$, in this case), such as ${\displaystyle {\hat {s}}[n]}$ or ${\displaystyle S[n]}$, and functional notation often replaces subscripting:

{\displaystyle {\begin{aligned}s_{\infty }(x)&=\sum _{n=-\infty }^{\infty }{\hat {s}}[n]\cdot e^{i\,2\pi nx/P}\\[6pt]&=\sum _{n=-\infty }^{\infty }S[n]\cdot e^{i\,2\pi nx/P}&&\scriptstyle {\mathsf {common\ engineering\ notation}}\end{aligned}}}

In engineering, particularly when the variable ${\displaystyle x}$ 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.

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

${\displaystyle S(f)\ \triangleq \ \sum _{n=-\infty }^{\infty }S[n]\cdot \delta \left(f-{\frac {n}{P}}\right),}$

where ${\displaystyle f}$ represents a continuous frequency domain. When variable ${\displaystyle x}$ has units of seconds, ${\displaystyle f}$ has units of hertz. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of ${\displaystyle 1/P}$, which is called the fundamental frequency.  ${\displaystyle s_{\infty }(x)}$  can be recovered from this representation by an inverse Fourier transform:

{\displaystyle {\begin{aligned}{\mathcal {F}}^{-1}\{S(f)\}&=\int _{-\infty }^{\infty }\left(\sum _{n=-\infty }^{\infty }S[n]\cdot \delta \left(f-{\frac {n}{P}}\right)\right)e^{i2\pi fx}\,df,\\[6pt]&=\sum _{n=-\infty }^{\infty }S[n]\cdot \int _{-\infty }^{\infty }\delta \left(f-{\frac {n}{P}}\right)e^{i2\pi fx}\,df,\\[6pt]&=\sum _{n=-\infty }^{\infty }S[n]\cdot e^{i\,2\pi nx/P}\ \ \triangleq \ s_{\infty }(x).\end{aligned}}}

The constructed function ${\displaystyle S(f)}$ 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. [upper-alpha 2]

Convergence

In engineering applications, the Fourier series is generally presumed to converge almost everywhere (the exceptions being at discrete discontinuities) since the functions encountered in engineering are better-behaved than the functions that mathematicians can provide as counter-examples to this presumption. In particular, if ${\displaystyle s}$ is continuous and the derivative of ${\displaystyle s(x)}$ (which may not exist everywhere) is square integrable, then the Fourier series of ${\displaystyle s}$ converges absolutely and uniformly to ${\displaystyle s(x)}$. [12] If a function is square-integrable on the interval ${\displaystyle [x_{0},x_{0}+P]}$, then the Fourier series converges to the function at almost every point. Convergence of Fourier series also depends on the finite number of maxima and minima in a function which is popularly known as one of the Dirichlet's condition for Fourier series. 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.

An interactive animation can be seen here.

Examples

Example 1: a simple Fourier series

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

${\displaystyle s(x)={\frac {x}{\pi }},\quad \mathrm {for} -\pi
${\displaystyle s(x+2\pi k)=s(x),\quad \mathrm {for} -\pi

In this case, the Fourier coefficients are given by

{\displaystyle {\begin{aligned}a_{n}&={\frac {1}{\pi }}\int _{-\pi }^{\pi }s(x)\cos(nx)\,dx=0,\quad n\geq 0.\\[4pt]b_{n}&={\frac {1}{\pi }}\int _{-\pi }^{\pi }s(x)\sin(nx)\,dx\\[4pt]&=-{\frac {2}{\pi n}}\cos(n\pi )+{\frac {2}{\pi ^{2}n^{2}}}\sin(n\pi )\\[4pt]&={\frac {2\,(-1)^{n+1}}{\pi n}},\quad n\geq 1.\end{aligned}}}

It can be proven that Fourier series converges to ${\displaystyle s(x)}$ at every point ${\displaystyle x}$ where ${\displaystyle s}$ is differentiable, and therefore:

{\displaystyle {\begin{aligned}s(x)&={\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }\left[a_{n}\cos \left(nx\right)+b_{n}\sin \left(nx\right)\right]\\[4pt]&={\frac {2}{\pi }}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}\sin(nx),\quad \mathrm {for} \quad x-\pi \notin 2\pi \mathbb {Z} .\end{aligned}}}

(Eq.7)

When ${\displaystyle x=\pi }$, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of s at ${\displaystyle x=\pi }$. This is a particular instance of the Dirichlet theorem for Fourier series.

This example leads us to a solution to the Basel problem.

Example 2: Fourier's motivation

The Fourier series expansion of our function in Example 1 looks more complicated than the simple formula ${\displaystyle s(x)=x/\pi }$, so it is not immediately apparent why one would need the Fourier series. While there are many applications, Fourier's motivation was in solving the heat equation. For example, consider a metal plate in the shape of a square whose sides measure ${\displaystyle \pi }$ meters, with coordinates ${\displaystyle (x,y)\in [0,\pi ]\times [0,\pi ]}$. 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 ${\displaystyle y=\pi }$, is maintained at the temperature gradient ${\displaystyle T(x,\pi )=x}$ degrees Celsius, for ${\displaystyle x}$ in ${\displaystyle (0,\pi )}$, then one can show that the stationary heat distribution (or the heat distribution after a long period of time has elapsed) is given by

${\displaystyle T(x,y)=2\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}\sin(nx){\sinh(ny) \over \sinh(n\pi )}.}$

Here, sinh is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of   Eq.7 by ${\displaystyle \sinh(ny)/\sinh(n\pi )}$. While our example function ${\displaystyle s(x)}$ seems to have a needlessly complicated Fourier series, the heat distribution ${\displaystyle T(x,y)}$ is nontrivial. The function ${\displaystyle T}$ cannot be written as a closed-form expression. This method of solving the heat problem was made possible by Fourier's work.

Example 3: complex Fourier series animation

An example of the ability of the complex Fourier series to draw any two dimensional closed figure is shown in the adjacent animation of the complex Fourier series converging to a drawing in the complex plane of the letter 'e' (for exponential). The animation alternates between fast rotations to take less time and slow rotations to show more detail. The terms of the complex Fourier series are shown in two rotating arms: one arm is an aggregate of all the complex Fourier series terms that rotate in the positive direction (counter clockwise, according to the right hand rule), the other arm is an aggregate of all the complex Fourier series terms that rotate in the negative direction. The constant term that does not rotate at all is evenly split between the two arms. The animation's small circle represents the midpoint between the extent of the two arms, which is also the midpoint between the origin and the complex Fourier series approximation which is the '+' symbol in the animation. (The GNU Octave source code for generating this animation is here. [13] Note that the animation uses the variable 't' to parameterize the drawing in the complex plane, equivalent to the use of the parameter 'x' in this article's subsection on complex valued functions.)

Other applications

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.

Beginnings

Joseph Fourier wrote:[ dubious ]

${\displaystyle \varphi (y)=a_{0}\cos {\frac {\pi y}{2}}+a_{1}\cos 3{\frac {\pi y}{2}}+a_{2}\cos 5{\frac {\pi y}{2}}+\cdots .}$

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

${\displaystyle a_{k}=\int _{-1}^{1}\varphi (y)\cos(2k+1){\frac {\pi y}{2}}\,dy.}$

This immediately gives any coefficient ak of the trigonometrical series for φ(y) for any function which has such an expansion. It works because if φ has such an expansion, then (under suitable convergence assumptions) the integral

{\displaystyle {\begin{aligned}a_{k}&=\int _{-1}^{1}\varphi (y)\cos(2k+1){\frac {\pi y}{2}}\,dy\\&=\int _{-1}^{1}\left(a\cos {\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}+a'\cos 3{\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}+\cdots \right)\,dy\end{aligned}}}

can be carried out term-by-term. But all terms involving ${\displaystyle \cos(2j+1){\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}}$ for jk 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 even rigour .[ citation needed ]

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, or for functions on a bounded (compact) interval.

Extensions

Fourier series on a square

We can also define the Fourier series for functions of two variables ${\displaystyle x}$ and ${\displaystyle y}$ in the square ${\displaystyle [-\pi ,\pi ]\times [-\pi ,\pi ]}$:

{\displaystyle {\begin{aligned}f(x,y)&=\sum _{j,k\in \mathbb {Z} }c_{j,k}e^{ijx}e^{iky},\\[5pt]c_{j,k}&={\frac {1}{4\pi ^{2}}}\int _{-\pi }^{\pi }\int _{-\pi }^{\pi }f(x,y)e^{-ijx}e^{-iky}\,dx\,dy.\end{aligned}}}

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-related transform using only the cosine basis functions.[ citation needed ]

Fourier series of Bravais-lattice-periodic-function

The three-dimensional Bravais lattice is defined as the set of vectors of the form:

${\displaystyle \mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}+n_{3}\mathbf {a} _{3}}$

where ${\displaystyle n_{i}}$ are integers and ${\displaystyle \mathbf {a} _{i}}$ are three linearly independent vectors. Assuming we have some function, ${\displaystyle f(\mathbf {r} )}$, such that it obeys the following condition for any Bravais lattice vector ${\displaystyle \mathbf {R} :f(\mathbf {r} )=f(\mathbf {R} +\mathbf {r} )}$, we could make a Fourier series of it. This kind of function can be, for example, the effective potential that one electron "feels" inside a periodic crystal. It is useful to make a Fourier series of the potential then when applying Bloch's theorem. First, we may write any arbitrary vector ${\displaystyle \mathbf {r} }$ in the coordinate-system of the lattice:

${\displaystyle \mathbf {r} =x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}},}$

where ${\displaystyle a_{i}\triangleq |\mathbf {a} _{i}|.}$

Thus we can define a new function,

${\displaystyle g(x_{1},x_{2},x_{3})\triangleq f(\mathbf {r} )=f\left(x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}}\right).}$

This new function, ${\displaystyle g(x_{1},x_{2},x_{3})}$, is now a function of three-variables, each of which has periodicity a1, a2, a3 respectively:

${\displaystyle g(x_{1},x_{2},x_{3})=g(x_{1}+a_{1},x_{2},x_{3})=g(x_{1},x_{2}+a_{2},x_{3})=g(x_{1},x_{2},x_{3}+a_{3}).}$

This enables us to build up a set of Fourier coefficients, each being indexed by three independent integers ${\displaystyle m_{1},m_{2},m_{3}}$. In what follows, we use function notation to denote these coefficients, where previously we used subscripts. If we write a series for g on the interval [0, a1] for x1, we can define the following:

${\displaystyle h^{\mathrm {one} }(m_{1},x_{2},x_{3})\triangleq {\frac {1}{a_{1}}}\int _{0}^{a_{1}}g(x_{1},x_{2},x_{3})\cdot e^{-i2\pi {\frac {m_{1}}{a_{1}}}x_{1}}\,dx_{1}}$

And then we can write:

${\displaystyle g(x_{1},x_{2},x_{3})=\sum _{m_{1}=-\infty }^{\infty }h^{\mathrm {one} }(m_{1},x_{2},x_{3})\cdot e^{i2\pi {\frac {m_{1}}{a_{1}}}x_{1}}}$

Further defining:

{\displaystyle {\begin{aligned}h^{\mathrm {two} }(m_{1},m_{2},x_{3})&\triangleq {\frac {1}{a_{2}}}\int _{0}^{a_{2}}h^{\mathrm {one} }(m_{1},x_{2},x_{3})\cdot e^{-i2\pi {\frac {m_{2}}{a_{2}}}x_{2}}\,dx_{2}\\[12pt]&={\frac {1}{a_{2}}}\int _{0}^{a_{2}}dx_{2}{\frac {1}{a_{1}}}\int _{0}^{a_{1}}dx_{1}g(x_{1},x_{2},x_{3})\cdot e^{-i2\pi \left({\frac {m_{1}}{a_{1}}}x_{1}+{\frac {m_{2}}{a_{2}}}x_{2}\right)}\end{aligned}}}

We can write ${\displaystyle g}$ once again as:

${\displaystyle g(x_{1},x_{2},x_{3})=\sum _{m_{1}=-\infty }^{\infty }\sum _{m_{2}=-\infty }^{\infty }h^{\mathrm {two} }(m_{1},m_{2},x_{3})\cdot e^{i2\pi {\frac {m_{1}}{a_{1}}}x_{1}}\cdot e^{i2\pi {\frac {m_{2}}{a_{2}}}x_{2}}}$

Finally applying the same for the third coordinate, we define:

{\displaystyle {\begin{aligned}h^{\mathrm {three} }(m_{1},m_{2},m_{3})&\triangleq {\frac {1}{a_{3}}}\int _{0}^{a_{3}}h^{\mathrm {two} }(m_{1},m_{2},x_{3})\cdot e^{-i2\pi {\frac {m_{3}}{a_{3}}}x_{3}}\,dx_{3}\\[12pt]&={\frac {1}{a_{3}}}\int _{0}^{a_{3}}dx_{3}{\frac {1}{a_{2}}}\int _{0}^{a_{2}}dx_{2}{\frac {1}{a_{1}}}\int _{0}^{a_{1}}dx_{1}g(x_{1},x_{2},x_{3})\cdot e^{-i2\pi \left({\frac {m_{1}}{a_{1}}}x_{1}+{\frac {m_{2}}{a_{2}}}x_{2}+{\frac {m_{3}}{a_{3}}}x_{3}\right)}\end{aligned}}}

We write ${\displaystyle g}$ as:

${\displaystyle g(x_{1},x_{2},x_{3})=\sum _{m_{1}=-\infty }^{\infty }\sum _{m_{2}=-\infty }^{\infty }\sum _{m_{3}=-\infty }^{\infty }h^{\mathrm {three} }(m_{1},m_{2},m_{3})\cdot e^{i2\pi {\frac {m_{1}}{a_{1}}}x_{1}}\cdot e^{i2\pi {\frac {m_{2}}{a_{2}}}x_{2}}\cdot e^{i2\pi {\frac {m_{3}}{a_{3}}}x_{3}}}$

Re-arranging:

${\displaystyle g(x_{1},x_{2},x_{3})=\sum _{m_{1},m_{2},m_{3}\in \mathbb {Z} }h^{\mathrm {three} }(m_{1},m_{2},m_{3})\cdot e^{i2\pi \left({\frac {m_{1}}{a_{1}}}x_{1}+{\frac {m_{2}}{a_{2}}}x_{2}+{\frac {m_{3}}{a_{3}}}x_{3}\right)}.}$

Now, every reciprocal lattice vector can be written as ${\displaystyle \mathbf {G} =\ell _{1}\mathbf {g} _{1}+\ell _{2}\mathbf {g} _{2}+\ell _{3}\mathbf {g} _{3}}$, where ${\displaystyle l_{i}}$ are integers and ${\displaystyle \mathbf {g} _{i}}$ are the reciprocal lattice vectors, we can use the fact that ${\displaystyle \mathbf {g_{i}} \cdot \mathbf {a_{j}} =2\pi \delta _{ij}}$ to calculate that for any arbitrary reciprocal lattice vector ${\displaystyle \mathbf {G} }$ and arbitrary vector in space ${\displaystyle \mathbf {r} }$, their scalar product is:

${\displaystyle \mathbf {G} \cdot \mathbf {r} =\left(\ell _{1}\mathbf {g} _{1}+\ell _{2}\mathbf {g} _{2}+\ell _{3}\mathbf {g} _{3}\right)\cdot \left(x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}}\right)=2\pi \left(x_{1}{\frac {\ell _{1}}{a_{1}}}+x_{2}{\frac {\ell _{2}}{a_{2}}}+x_{3}{\frac {\ell _{3}}{a_{3}}}\right).}$

And so it is clear that in our expansion, the sum is actually over reciprocal lattice vectors:

${\displaystyle f(\mathbf {r} )=\sum _{\mathbf {G} }h(\mathbf {G} )\cdot e^{i\mathbf {G} \cdot \mathbf {r} },}$

where

${\displaystyle h(\mathbf {G} )={\frac {1}{a_{3}}}\int _{0}^{a_{3}}dx_{3}\,{\frac {1}{a_{2}}}\int _{0}^{a_{2}}dx_{2}\,{\frac {1}{a_{1}}}\int _{0}^{a_{1}}dx_{1}\,f\left(x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}}\right)\cdot e^{-i\mathbf {G} \cdot \mathbf {r} }.}$

Assuming

${\displaystyle \mathbf {r} =(x,y,z)=x_{1}{\frac {\mathbf {a} _{1}}{a_{1}}}+x_{2}{\frac {\mathbf {a} _{2}}{a_{2}}}+x_{3}{\frac {\mathbf {a} _{3}}{a_{3}}},}$

we can solve this system of three linear equations for ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ in terms of ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$ and ${\displaystyle x_{3}}$ in order to calculate the volume element in the original cartesian coordinate system. Once we have ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$ in terms of ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$ and ${\displaystyle x_{3}}$, we can calculate the Jacobian determinant:

${\displaystyle {\begin{vmatrix}{\dfrac {\partial x_{1}}{\partial x}}&{\dfrac {\partial x_{1}}{\partial y}}&{\dfrac {\partial x_{1}}{\partial z}}\\[12pt]{\dfrac {\partial x_{2}}{\partial x}}&{\dfrac {\partial x_{2}}{\partial y}}&{\dfrac {\partial x_{2}}{\partial z}}\\[12pt]{\dfrac {\partial x_{3}}{\partial x}}&{\dfrac {\partial x_{3}}{\partial y}}&{\dfrac {\partial x_{3}}{\partial z}}\end{vmatrix}}}$

which after some calculation and applying some non-trivial cross-product identities can be shown to be equal to:

${\displaystyle {\frac {a_{1}a_{2}a_{3}}{\mathbf {a} _{1}\cdot (\mathbf {a} _{2}\times \mathbf {a} _{3})}}}$

(it may be advantageous for the sake of simplifying calculations, to work in such a cartesian coordinate system, in which it just so happens that ${\displaystyle \mathbf {a} _{1}}$ is parallel to the x axis, ${\displaystyle \mathbf {a} _{2}}$ lies in the xy-plane, and ${\displaystyle \mathbf {a} _{3}}$ has components of all three axes). The denominator is exactly the volume of the primitive unit cell which is enclosed by the three primitive-vectors ${\displaystyle \mathbf {a} _{1}}$, ${\displaystyle \mathbf {a} _{2}}$ and ${\displaystyle \mathbf {a} _{3}}$. In particular, we now know that

${\displaystyle dx_{1}\,dx_{2}\,dx_{3}={\frac {a_{1}a_{2}a_{3}}{\mathbf {a} _{1}\cdot (\mathbf {a} _{2}\times \mathbf {a} _{3})}}\cdot dx\,dy\,dz.}$

We can write now ${\displaystyle h(\mathbf {G} )}$ as an integral with the traditional coordinate system over the volume of the primitive cell, instead of with the ${\displaystyle x_{1}}$, ${\displaystyle x_{2}}$ and ${\displaystyle x_{3}}$ variables:

${\displaystyle h(\mathbf {G} )={\frac {1}{\mathbf {a} _{1}\cdot (\mathbf {a} _{2}\times \mathbf {a} _{3})}}\int _{C}d\mathbf {r} f(\mathbf {r} )\cdot e^{-i\mathbf {G} \cdot \mathbf {r} }}$

writing ${\displaystyle d\mathbf {r} }$ for the volume element ${\displaystyle dx\,dy\,dz}$; and where ${\displaystyle C}$ is the primitive unit cell, thus, ${\displaystyle \mathbf {a} _{1}\cdot (\mathbf {a} _{2}\times \mathbf {a} _{3})}$ is the volume of the primitive unit cell.

Hilbert space interpretation

In the language of Hilbert spaces, the set of functions ${\displaystyle \left\{e_{n}=e^{inx}:n\in \mathbb {Z} \right\}}$ is an orthonormal basis for the space ${\displaystyle L^{2}([-\pi ,\pi ])}$ of square-integrable functions on ${\displaystyle [-\pi ,\pi ]}$. This space is actually a Hilbert space with an inner product given for any two elements ${\displaystyle f}$ and ${\displaystyle g}$ by:

${\displaystyle \langle f,\,g\rangle \;\triangleq \;{\frac {1}{2\pi }}\int _{-\pi }^{\pi }f(x)g^{*}(x)\,dx,}$   where ${\displaystyle g^{*}(x)}$ is the complex conjugate of ${\displaystyle g(x).}$

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

${\displaystyle f=\sum _{n=-\infty }^{\infty }\langle f,e_{n}\rangle \,e_{n}.}$

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:

${\displaystyle \int _{-\pi }^{\pi }\cos(mx)\,\cos(nx)\,dx={\frac {1}{2}}\int _{-\pi }^{\pi }\cos((n-m)x)+\cos((n+m)x)\,dx=\pi \delta _{mn},\quad m,n\geq 1,}$
${\displaystyle \int _{-\pi }^{\pi }\sin(mx)\,\sin(nx)\,dx={\frac {1}{2}}\int _{-\pi }^{\pi }\cos((n-m)x)-\cos((n+m)x)\,dx=\pi \delta _{mn},\quad m,n\geq 1}$

(where δmn is the Kronecker delta), and

${\displaystyle \int _{-\pi }^{\pi }\cos(mx)\,\sin(nx)\,dx={\frac {1}{2}}\int _{-\pi }^{\pi }\sin((n+m)x)+\sin((n-m)x)\,dx=0;\,}$

furthermore, the sines and cosines are orthogonal to the constant function ${\displaystyle 1}$. An orthonormal basis for ${\displaystyle L^{2}([-\pi ,\pi ])}$ consisting of real functions is formed by the functions ${\displaystyle 1}$ and ${\displaystyle {\sqrt {2}}\cos(nx)}$, ${\displaystyle {\sqrt {2}}\sin(nx)}$ with 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

Table of basic properties

This table shows some mathematical operations in the time domain and the corresponding effect in the Fourier series coefficients. Notation:

• Complex conjugation is denoted by an asterisk.
• ${\displaystyle f(x),g(x)}$ designate ${\displaystyle P}$-periodic functions or functions defined only for ${\displaystyle x\in [0,P]}$.
• ${\displaystyle F[n],G[n]}$ designate the Fourier series coefficients (exponential form) of ${\displaystyle f}$ and ${\displaystyle g}$ as defined in equation Eq.5 .
PropertyTime domainFrequency domain (exponential form)RemarksReference
Linearity${\displaystyle a\cdot f(x)+b\cdot g(x)}$${\displaystyle a\cdot F[n]+b\cdot G[n]}$${\displaystyle a,b\in \mathbb {C} }$
Time reversal / Frequency reversal${\displaystyle f(-x)}$${\displaystyle F[-n]}$ [15] :p. 610
Time conjugation${\displaystyle f^{*}(x)}$${\displaystyle F^{*}[-n]}$ [15] :p. 610
Time reversal & conjugation${\displaystyle f^{*}(-x)}$${\displaystyle F^{*}[n]}$
Real part in time${\displaystyle \operatorname {Re} {(f(x))}}$${\displaystyle {\frac {1}{2}}(F[n]+F^{*}[-n])}$
Imaginary part in time${\displaystyle \operatorname {Im} {(f(x))}}$${\displaystyle {\frac {1}{2i}}(F[n]-F^{*}[-n])}$
Real part in frequency${\displaystyle {\frac {1}{2}}(f(x)+f^{*}(-x))}$${\displaystyle \operatorname {Re} {(F[n])}}$
Imaginary part in frequency${\displaystyle {\frac {1}{2i}}(f(x)-f^{*}(-x))}$${\displaystyle \operatorname {Im} {(F[n])}}$
Shift in time / Modulation in frequency${\displaystyle f(x-x_{0})}$${\displaystyle F[n]\cdot e^{-i{\frac {2\pi x_{0}}{P}}n}}$${\displaystyle x_{0}\in \mathbb {R} }$ [15] :p. 610
Shift in frequency / Modulation in time${\displaystyle f(x)\cdot e^{i{\frac {2\pi n_{0}}{P}}x}}$${\displaystyle F[n-n_{0}]\!}$${\displaystyle n_{0}\in \mathbb {Z} }$ [15] :p. 610

Symmetry properties

When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform: [16]

${\displaystyle {\begin{array}{rccccccccc}{\text{Time domain}}&f&=&f_{_{\text{RE}}}&+&f_{_{\text{RO}}}&+&if_{_{\text{IE}}}&+&\underbrace {i\ f_{_{\text{IO}}}} \\&{\Bigg \Updownarrow }{\mathcal {F}}&&{\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}&&\ \ {\Bigg \Updownarrow }{\mathcal {F}}\\{\text{Frequency domain}}&F&=&F_{RE}&+&\overbrace {i\ F_{IO}} &+&i\ F_{IE}&+&F_{RO}\end{array}}}$

From this, various relationships are apparent, for example:

• The transform of a real-valued function (fRE+ fRO) is the even symmetric function FRE+ i FIO. Conversely, an even-symmetric transform implies a real-valued time-domain.
• The transform of an imaginary-valued function (ifIE+ ifIO) is the odd symmetric function FRO+ i FIE, and the converse is true.
• The transform of an even-symmetric function (fRE+ ifIO) is the real-valued function FRE+ FRO, and the converse is true.
• The transform of an odd-symmetric function (fRO+ ifIE) is the imaginary-valued function i FIE+ i FIO, and the converse is true.

Riemann–Lebesgue lemma

If ${\displaystyle f}$ is integrable, ${\textstyle \lim _{|n|\to \infty }F[n]=0}$, ${\textstyle \lim _{n\to +\infty }a_{n}=0}$ and ${\textstyle \lim _{n\to +\infty }b_{n}=0.}$ This result is known as the Riemann–Lebesgue lemma.

Parseval's theorem

If ${\displaystyle f}$ belongs to ${\displaystyle L^{2}(P)}$ (an interval of length ${\displaystyle P}$) then:${\textstyle \sum _{n=-\infty }^{\infty }{\Bigl |}F[n]{\Bigr |}^{2}={\frac {1}{P}}\int _{P}|f(x)|^{2}\,dx.}$

Plancherel's theorem

If ${\displaystyle c_{0},\,c_{\pm 1},\,c_{\pm 2},\ldots }$ are coefficients and ${\textstyle \sum _{n=-\infty }^{\infty }|c_{n}|^{2}<\infty }$ then there is a unique function ${\displaystyle f\in L^{2}(P)}$ such that ${\displaystyle F[n]=c_{n}}$ for every ${\displaystyle n}$.

Convolution theorems

Given ${\displaystyle P}$-periodic functions, ${\displaystyle f_{_{P}}}$ and ${\displaystyle g_{_{P}}}$ with Fourier series coefficients ${\displaystyle F[n]}$ and ${\displaystyle G[n],}$${\displaystyle n\in \mathbb {Z} ,}$

• The pointwise product:
${\displaystyle h_{_{P}}(x)\triangleq f_{_{P}}(x)\cdot g_{_{P}}(x)}$
is also ${\displaystyle P}$-periodic, and its Fourier series coefficients are given by the discrete convolution of the ${\displaystyle F}$ and ${\displaystyle G}$ sequences:
${\displaystyle H[n]=\{F*G\}[n].}$
• The periodic convolution :
${\displaystyle h_{_{P}}(x)\triangleq \int _{P}f_{_{P}}(\tau )\cdot g_{_{P}}(x-\tau )\,d\tau }$
is also ${\displaystyle P}$-periodic, with Fourier series coefficients:
${\displaystyle H[n]=P\cdot F[n]\cdot G[n].}$
• A doubly infinite sequence ${\displaystyle \left\{c_{n}\right\}_{n\in Z}}$ in ${\displaystyle c_{0}(\mathbb {Z} )}$ is the sequence of Fourier coefficients of a function in ${\displaystyle L^{1}([0,2\pi ])}$ if and only if it is a convolution of two sequences in ${\displaystyle \ell ^{2}(\mathbb {Z} )}$. See [17]

Derivative property

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

• If ${\displaystyle f\in C^{1}(\mathbb {T} )}$, then the Fourier coefficients ${\displaystyle {\widehat {f'}}[n]}$ of the derivative ${\displaystyle f'}$ can be expressed in terms of the Fourier coefficients ${\displaystyle {\widehat {f}}[n]}$ of the function ${\displaystyle f}$, via the formula ${\displaystyle {\widehat {f'}}[n]=in{\widehat {f}}[n]}$.
• If ${\displaystyle f\in C^{k}(\mathbb {T} )}$, then ${\displaystyle {\widehat {f^{(k)}}}[n]=(in)^{k}{\widehat {f}}[n]}$. In particular, since for a fixed ${\displaystyle k\geq 1}$ we have ${\displaystyle {\widehat {f^{(k)}}}[n]\to 0}$ as ${\displaystyle n\to \infty }$, it follows that ${\displaystyle |n|^{k}{\widehat {f}}[n]}$ tends to zero, which means that the Fourier coefficients converge to zero faster than the kth power of n for any ${\displaystyle k\geq 1}$.

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 [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.

Riemannian manifolds

If the domain is not a group, then there is no intrinsically defined convolution. However, if ${\displaystyle X}$ 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 ${\displaystyle X}$. Then, by analogy, one can consider heat equations on ${\displaystyle X}$. 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 ${\displaystyle L^{2}(X)}$, where ${\displaystyle X}$ is a Riemannian manifold. The Fourier series converges in ways similar to the ${\displaystyle [-\pi ,\pi ]}$ case. A typical example is to take ${\displaystyle X}$ 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 straightforward generalization to Locally Compact Abelian (LCA) groups.

This generalizes the Fourier transform to ${\displaystyle L^{1}(G)}$ or ${\displaystyle L^{2}(G)}$, where ${\displaystyle G}$ is an LCA group. If ${\displaystyle G}$ is compact, one also obtains a Fourier series, which converges similarly to the ${\displaystyle [-\pi ,\pi ]}$ case, but if ${\displaystyle G}$ is noncompact, one obtains instead a Fourier integral. This generalization yields the usual Fourier transform when the underlying locally compact Abelian group is ${\displaystyle \mathbb {R} }$.

Table of common Fourier series

Some common pairs of periodic functions and their Fourier Series coefficients are shown in the table below. The following notation applies:

• ${\displaystyle f(x)}$ designates a periodic function defined on ${\displaystyle 0.
• ${\displaystyle a_{0},a_{n},b_{n}}$ designate the Fourier Series coefficients (sine-cosine form) of the periodic function ${\displaystyle f}$ as defined in Eq.1 .
Time domain
${\displaystyle f(x)}$
PlotFrequency domain (sine-cosine form)
{\displaystyle {\begin{aligned}&a_{0}\\&a_{n}\quad {\text{for }}n\geq 1\\&b_{n}\quad {\text{for }}n\geq 1\end{aligned}}}
RemarksReference
${\displaystyle f(x)=A\left|\sin \left({\frac {2\pi }{P}}x\right)\right|\quad {\text{for }}0\leq x{\displaystyle {\begin{aligned}a_{0}=&{\frac {4A}{\pi }}\\a_{n}=&{\begin{cases}{\frac {-4A}{\pi }}{\frac {1}{n^{2}-1}}&\quad n{\text{ even}}\\0&\quad n{\text{ odd}}\end{cases}}\\b_{n}=&0\\\end{aligned}}}Full-wave rectified sine [18] :p. 193
${\displaystyle f(x)={\begin{cases}A\sin \left({\frac {2\pi }{P}}x\right)&\quad {\text{for }}0\leq x{\displaystyle {\begin{aligned}a_{0}=&{\frac {2A}{\pi }}\\a_{n}=&{\begin{cases}{\frac {-2A}{\pi }}{\frac {1}{n^{2}-1}}&\quad n{\text{ even}}\\0&\quad n{\text{ odd}}\end{cases}}\\b_{n}=&{\begin{cases}{\frac {A}{2}}&\quad n=1\\0&\quad n>1\end{cases}}\\\end{aligned}}}Half-wave rectified sine [18] :p. 193
${\displaystyle f(x)={\begin{cases}A&\quad {\text{for }}0\leq x{\displaystyle {\begin{aligned}a_{0}=&2AD\\a_{n}=&{\frac {A}{n\pi }}\sin \left(2\pi nD\right)\\b_{n}=&{\frac {2A}{n\pi }}\left(\sin \left(\pi nD\right)\right)^{2}\\\end{aligned}}}${\displaystyle 0\leq D\leq 1}$
${\displaystyle f(x)={\frac {Ax}{P}}\quad {\text{for }}0\leq x{\displaystyle {\begin{aligned}a_{0}=&A\\a_{n}=&0\\b_{n}=&{\frac {-A}{n\pi }}\\\end{aligned}}} [18] :p. 192
${\displaystyle f(x)=A-{\frac {Ax}{P}}\quad {\text{for }}0\leq x{\displaystyle {\begin{aligned}a_{0}=&A\\a_{n}=&0\\b_{n}=&{\frac {A}{n\pi }}\\\end{aligned}}} [18] :p. 192
${\displaystyle f(x)={\frac {4A}{P^{2}}}\left(x-{\frac {P}{2}}\right)^{2}\quad {\text{for }}0\leq x{\displaystyle {\begin{aligned}a_{0}=&{\frac {2A}{3}}\\a_{n}=&{\frac {4A}{\pi ^{2}n^{2}}}\\b_{n}=&0\\\end{aligned}}} [18] :p. 193

Approximation and convergence of Fourier series

Recalling Eq.5 ,

${\displaystyle s_{_{N}}(x)=\sum _{n=-N}^{N}S[n]\ e^{i{\tfrac {2\pi }{P}}nx},}$

it is a trigonometric polynomial of degree ${\displaystyle N}$, generally:

${\displaystyle p_{_{N}}(x)=\sum _{n=-N}^{N}p[n]\ e^{i{\tfrac {2\pi }{P}}nx}.}$

Least squares property

Parseval's theorem implies that:

Theorem. The trigonometric polynomial ${\displaystyle s_{_{N}}}$ is the unique best trigonometric polynomial of degree ${\displaystyle N}$ approximating ${\displaystyle s(x)}$, in the sense that, for any trigonometric polynomial ${\displaystyle p_{_{N}}\neq s_{_{N}}}$ of degree ${\displaystyle N}$, we have:

${\displaystyle \|s_{_{N}}-s\|_{2}<\|p_{_{N}}-s\|_{2},}$

where the Hilbert space norm is defined as:

${\displaystyle \|g\|_{2}={\sqrt {{1 \over P}\int _{P}|g(x)|^{2}\,dx}}.}$

Convergence

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

Theorem. If ${\displaystyle s}$ belongs to ${\displaystyle L^{2}(P)}$ (an interval of length ${\displaystyle P}$), then ${\displaystyle s_{\infty }}$ converges to ${\displaystyle s}$ in ${\displaystyle L^{2}(P)}$, that is,${\displaystyle \|s_{_{N}}-s\|_{2}}$ converges to 0 as ${\displaystyle N\rightarrow \infty }$.

We have already mentioned that if ${\displaystyle s}$ is continuously differentiable, then  ${\displaystyle (i\cdot n)S[n]}$  is the nth Fourier coefficient of the derivative ${\displaystyle s'}$. It follows, essentially from the Cauchy–Schwarz inequality, that ${\displaystyle s_{\infty }}$ is absolutely summable. The sum of this series is a continuous function, equal to ${\displaystyle s}$, since the Fourier series converges in the mean to ${\displaystyle s}$:

Theorem. If ${\displaystyle s\in C^{1}(\mathbb {T} )}$, then ${\displaystyle s_{\infty }}$ converges to ${\displaystyle s}$ uniformly (and hence also pointwise.)

This result can be proven easily if ${\displaystyle s}$ is further assumed to be ${\displaystyle C^{2}}$, since in that case ${\displaystyle n^{2}S[n]}$ tends to zero as ${\displaystyle n\rightarrow \infty }$. More generally, the Fourier series is absolutely summable, thus converges uniformly to ${\displaystyle s}$, provided that ${\displaystyle s}$ satisfies a Hölder condition of order ${\displaystyle \alpha >1/2}$. In the absolutely summable case, the inequality:

${\displaystyle \sup _{x}|s(x)-s_{_{N}}(x)|\leq \sum _{|n|>N}|S[n]|}$  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 ${\displaystyle x}$ if ${\displaystyle s}$ is differentiable at ${\displaystyle x}$, to Lennart Carleson's much more sophisticated result that the Fourier series of an ${\displaystyle L^{2}}$ 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". [19] [20] [21] [22]

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.[ citation needed ] The uniform boundedness principle yields a simple non-constructive proof of this fact.

In 1922, Andrey Kolmogorov published an article titled 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 ).

Notes

1. These three did some important early work on the wave equation, especially D'Alembert. Euler's work in this area was mostly Euler–Bernoulli beam equation|comtemporaneous/ in collaboration with Bernoulli, although the latter made some independent contributions to the theory of waves and vibrations. (See Fetter & Walecka 2003, pp. 209–210).
2. 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 ${\displaystyle {\mathcal {F}}\{e^{i{\frac {2\pi nx}{P}}}\}}$ is a Dirac delta function, which is an example of a distribution.
3. These words are not strictly Fourier's. Whilst the cited article does list the author as Fourier, a footnote indicates that the article was actually written by Poisson (that it was not written by Fourier is also clear from the consistent use of the third person to refer to him) and that it is, "for reasons of historical interest", presented as though it were Fourier's original memoire.

Related Research Articles

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies. It has the same sample-values as the original input sequence. The DFT is therefore said to be a frequency domain representation of the original input sequence. If the original sequence spans all the non-zero values of a function, its DTFT is continuous, and the DFT provides discrete samples of one cycle. If the original sequence is one cycle of a periodic function, the DFT provides all the non-zero values of one DTFT cycle.

In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor, who introduced them in 1715.

In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as

In mathematics, the Dirac delta function is a generalized function or distribution, a function on the space of test functions. It was introduced by physicist Paul Dirac. It is called a function, although it is not a function RC.

In mathematics, a Fourier transform (FT) is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial or temporal frequency, such as the expression of a musical chord in terms of the volumes and frequencies of its constituent notes. The term Fourier transform refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time.

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be thought of as an integral version of the product rule of differentiation.

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis is called an orthonormal basis.

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the form

In mathematics, the Gibbs phenomenon, discovered by Henry Wilbraham (1848) and rediscovered by J. Willard Gibbs (1899), is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself. The overshoot does not die out as n increases, but approaches a finite limit. This sort of behavior was also observed by experimental physicists, but was believed to be due to imperfections in the measuring apparatus.

In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 by Johann Radon, who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes. It was later generalized to higher-dimensional Euclidean spaces, and more broadly in the context of integral geometry. The complex analogue of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object.

In mathematics, physics and engineering, the sinc function, denoted by sinc(x), has two forms, normalized and unnormalized.

In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series. It is a non-negative kernel, giving rise to an approximate identity. It is named after the Hungarian mathematician Lipót Fejér (1880–1959).

In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum of the square of a function is equal to the sum of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh.

In mathematics, the Hankel transform expresses any given function f(r) as the weighted sum of an infinite number of Bessel functions of the first kind Jν(kr). The Bessel functions in the sum are all of the same order ν, but differ in a scaling factor k along the r axis. The necessary coefficient Fν of each Bessel function in the sum, as a function of the scaling factor k constitutes the transformed function. The Hankel transform is an integral transform and was first developed by the mathematician Hermann Hankel. It is also known as the Fourier–Bessel transform. Just as the Fourier transform for an infinite interval is related to the Fourier series over a finite interval, so the Hankel transform over an infinite interval is related to the Fourier–Bessel series over a finite interval.

In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.

Common integrals in quantum field theory are all variations and generalizations of Gaussian integrals to the complex plane and to multiple dimensions. Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered.

In optics, the Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.

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