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Fourier Series
Decomposition of periodic functions into sums of simpler sinusoidal forms
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.
Function $s(x)$ (in red) is a sum of six sine functions of different amplitudes and harmonically related frequencies. Their summation is called a Fourier series. The Fourier transform, $S(f)$ (in blue), which depicts amplitude vs frequency, reveals the 6 frequencies (at odd harmonics) and their amplitudes (1/odd number).
History
The Fourier series is named in honour 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.^{[A]} 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 function^{[3]} 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, $s(x)$, that is integrable on an interval of length $P$, which will be the period of the Fourier series. Common examples of analysis intervals are:
$x\in [0,1],$ and $P=1.$
$x\in [-\pi ,\pi ],$ and $P=2\pi .$
The analysis process determines the weights, indexed by integer $n$, which is also the number of cycles of the $n^{\text{th}}$ harmonic in the analysis interval. Therefore, the length of a cycle, in the units of $x$, is $P/n$. And the corresponding harmonic frequency is $n/P$. The $n^{th}$ harmonics are $\sin \left(2\pi x{\tfrac {n}{P}}\right)$ and $\cos \left(2\pi x{\tfrac {n}{P}}\right)$, and their amplitudes (weights) are found by integration over the interval of length $P$:^{[11]}
Fourier coefficients
If $s(x)$ is $P$-periodic, then any interval of that length is sufficient.
$a_{0}$ and $b_{0}$ can be reduced to $a_{0}={\frac {2}{P}}\int _{P}s(x)\,dx$ and $b_{0}=0$.
Many texts choose $P=2\pi$ to simplify the argument of the sinusoid functions.
The synthesis process (the actual Fourier series) is:
Fourier series, sine-cosine form
In general, integer $N$ is theoretically infinite. Even so, the series might not converge or exactly equate to $s(x)$ at all values of $x$ (such as a single-point discontinuity) in the analysis interval. For the "well-behaved" functions typical of physical processes, equality is customarily assumed.
If $s(t)$ is a function contained in an interval of length $P$ (and zero elsewhere), the upper-right quadrant is an example of what its Fourier series coefficients ($A_{n}$) might look like when plotted against their corresponding harmonic frequencies. The upper-left quadrant is the corresponding Fourier transform of $s(t).$ The Fourier series summation (not shown) synthesizes a periodic summation of $s(t),$ whereas the inverse Fourier transform (not shown) synthesizes only $s(t).$
and definitions $A_{n}\triangleq {\sqrt {a_{n}^{2}+b_{n}^{2}}}$ and $\varphi _{n}\triangleq \operatorname {arctan2} (b_{n},a_{n})$,
the sine and cosine pairs can be expressed as a single sinusoid with a phase offset, analogous to the conversion between orthogonal (Cartesian) and polar coordinates:
Fourier series, amplitude-phase form
The customary form for generalizing to complex-valued $s(x)$ (next section) is obtained using Euler's formula to split the cosine function into complex exponentials. Here, complex conjugation is denoted by an asterisk:
If $s(x)$ is a complex-valued function of a real variable $x,$ 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:
The notation $c_{n}$ is inadequate for discussing the Fourier coefficients of several different functions. Therefore, it is customarily replaced by a modified form of the function ($s$, in this case), such as ${\hat {s}}(n)$ or $S[n]$, 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.
Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb:
where $f$ represents a continuous frequency domain. When variable $x$ has units of seconds, $f$ has units of hertz. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of $1/P$, which is called the fundamental frequency. $s_{\infty }(x)$ can be recovered from this representation by an inverse Fourier transform:
The constructed function $S(f)$ 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.^{[B]}
The first four partial sums of the Fourier series for a square wave
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 $s$ is continuous and the derivative of $s(x)$ (which may not exist everywhere) is square integrable, then the Fourier series of $s$ converges absolutely and uniformly to $s(x)$.^{[12]} If a function is square-integrable on the interval $[x_{0},x_{0}+P]$, 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.
Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a square wave improves as the number of terms increases (animation)
Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a sawtooth wave improves as the number of terms increases (animation)
Example of convergence to a somewhat arbitrary function. Note the development of the "ringing" (Gibbs phenomenon) at the transitions to/from the vertical sections.
It can be proven that Fourier series converges to $s(x)$ at every point $x$ where $s$ is differentiable, and therefore:
When $x=\pi$, the Fourier series converges to 0, which is the half-sum of the left- and right-limit of s at $x=\pi$. 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
Heat distribution in a metal plate, using Fourier's method
The Fourier series expansion of our function in Example 1 looks more complicated than the simple formula $s(x)=x/\pi$, 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 $\pi$ meters, with coordinates $(x,y)\in [0,\pi ]\times [0,\pi ]$. 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 $y=\pi$, is maintained at the temperature gradient $T(x,\pi )=x$ degrees Celsius, for $x$ in $(0,\pi )$, then one can show that the stationary heat distribution (or the heat distribution after a long period of time has elapsed) is given by
Here, sinh is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of Eq.7 by $\sinh(ny)/\sinh(n\pi )$. While our example function $s(x)$ 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.
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 ?(2n), for any positive integer n.
This immediately gives any coefficient a_{k} of the trigonometrical series for ?(y) 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 $\cos(2j+1){\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}$ for 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.^{[]}
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 $x$ and $y$ in the square $[-\pi ,\pi ]\times [-\pi ,\pi ]$:
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.^{[]}
Fourier series of Bravais-lattice-periodic-function
The three-dimensional Bravais lattice is defined as the set of vectors of the form:
where $n_{i}$ are integers and $\mathbf {a} _{i}$ are three linearly independent vectors. Assuming we have some function, $f(\mathbf {r} )$, such that it obeys the following condition for any Bravais lattice vector $\mathbf {R} :f(\mathbf {r} )=f(\mathbf {R} +\mathbf {r} )$, 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 $\mathbf {r}$ in the coordinate-system of the lattice:
This enables us to build up a set of Fourier coefficients, each being indexed by three independent integers $m_{1},m_{2},m_{3}$. In what follows, we use function notation to denote these coefficents, where previously we used subscripts.
If we write a series for g on the interval [0, a_{1}] for x_{1}, we can define the following:
Now, every reciprocal lattice vector can be written as $\mathbf {G} =\ell _{1}\mathbf {g} _{1}+\ell _{2}\mathbf {g} _{2}+\ell _{3}\mathbf {g} _{3}$, where $l_{i}$ are integers and $\mathbf {g} _{i}$ are the reciprocal lattice vectors, we can use the fact that $\mathbf {g_{i}} \cdot \mathbf {a_{j}} =2\pi \delta _{ij}$ to calculate that for any arbitrary reciprocal lattice vector $\mathbf {G}$ and arbitrary vector in space $\mathbf {r}$, their scalar product is:
we can solve this system of three linear equations for $x$, $y$, and $z$ in terms of $x_{1}$, $x_{2}$ and $x_{3}$ in order to calculate the volume element in the original cartesian coordinate system. Once we have $x$, $y$, and $z$ in terms of $x_{1}$, $x_{2}$ and $x_{3}$, we can calculate the Jacobian determinant:
(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 $\mathbf {a_{1}}$ is parallel to the x axis, $\mathbf {a_{2}}$ lies in the x-y plane, and $\mathbf {a_{3}}$ 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 $\mathbf {a_{1}}$, $\mathbf {a_{2}}$ and $\mathbf {a_{3}}$. In particular, we now know that
We can write now $h(\mathbf {G} )$ as an integral with the traditional coordinate system over the volume of the primitive cell, instead of with the $x_{1}$, $x_{2}$ and $x_{3}$ variables:
writing $d\mathbf {r}$ for the volume element $dx\,dy\,dz$; and where $C$ is the primitive unit cell, thus, $\mathbf {a_{1}} \cdot (\mathbf {a_{2}} \times \mathbf {a_{3}} )$ is the volume of the primitive unit cell.
Hilbert space interpretation
In the language of Hilbert spaces, the set of functions $\{e_{n}=e^{inx}:n\in \mathbb {Z} \}$ is an orthonormal basis for the space $L^{2}([-\pi ,\pi ])$ of square-integrable functions on $[-\pi ,\pi ]$. This space is actually a Hilbert space with an inner product given for any two elements $f$ and $g$ by
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 $m$, $n$ or the functions are different, and pi only if $m$ and $n$ are equal, and the function used is the same.
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:
furthermore, the sines and cosines are orthogonal to the constant function $1$. An orthonormal basis for $L^{2}([-\pi ,\pi ])$ consisting of real functions is formed by the functions $1$ and ${\sqrt {2}}\cos(nx)$, ${\sqrt {2}}\sin(nx)$ 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:
$f(x),g(x)$ designate $P$-periodic functions defined on $0<x\leq P$.
$F[n],G[n]$ designate the Fourier series coefficients (exponential form) of $f$ and $g$ as defined in equation Eq.5.
Property
Time domain
Frequency domain (exponential form)
Remarks
Reference
Linearity
$a\cdot f(x)+b\cdot g(x)$
$a\cdot F[n]+b\cdot G[n]$
complex numbers $a,b$
Time reversal / Frequency reversal
$f(-x)$
$F[-n]$
^{[14]}^{:p. 610}
Time conjugation
$f(x)^{*}$
$F[-n]^{*}$
^{[14]}^{:p. 610}
Time reversal & conjugation
$f(-x)^{*}$
$F[n]^{*}$
Real part in time
$\operatorname {Re} {(f(x))}$
${\frac {1}{2}}(F[n]+F[-n]^{*})$
Imaginary part in time
$\operatorname {Im} {(f(x))}$
${\frac {1}{2i}}(F[n]-F[-n]^{*})$
Real part in frequency
${\frac {1}{2}}(f(x)+f(-x)^{*})$
$\operatorname {Re} {(F[n])}$
Imaginary part in frequency
${\frac {1}{2i}}(f(x)-f(-x)^{*})$
$\operatorname {Im} {(F[n])}$
Shift in time / Modulation in frequency
$f(x-x_{0})$
$F[n]\cdot e^{-i{\frac {2\pi x_{0}}{T}}n}$
real number $x_{0}$
^{[14]}^{:p. 610}
Shift in frequency / Modulation in time
$f(x)\cdot e^{i{\frac {2\pi n_{0}}{T}}x}$
$F[n-n_{0}]\!$
integer $n_{0}$
^{[14]}^{: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:^{[15]}
From this, various relationships are apparent, for example:
The transform of a real-valued function (f_{RE}+ f_{RO}) is the even symmetric function F_{RE}+ i F_{IO}. Conversely, an even-symmetric transform implies a real-valued time-domain.
The transform of an imaginary-valued function (if_{IE}+ if_{IO}) is the odd symmetric function F_{RO}+ i F_{IE}, and the converse is true.
The transform of an even-symmetric function (f_{RE}+ if_{IO}) is the real-valued function F_{RE}+ F_{RO}, and the converse is true.
The transform of an odd-symmetric function (f_{RO}+ if_{IE}) is the imaginary-valued function i F_{IE}+ i F_{IO}, and the converse is true.
Riemann-Lebesgue lemma
If $f$ is integrable, $\lim _{|n|\rightarrow \infty }{\hat {f}}(n)=0$, $\lim _{n\rightarrow +\infty }a_{n}=0$ and $\lim _{n\rightarrow +\infty }b_{n}=0.$ This result is known as the Riemann-Lebesgue lemma.
Derivative property
We say that $f$ belongs to
$C^{k}(\mathbb {T} )$ if $f$ is a 2?-periodic function on $\mathbb {R}$ which is $k$ times differentiable, and its kth derivative is continuous.
If $f\in C^{1}(\mathbb {T} )$, then the Fourier coefficients ${\widehat {f'}}(n)$ of the derivative $f'$ can be expressed in terms of the Fourier coefficients ${\widehat {f}}(n)$ of the function $f$, via the formula ${\widehat {f'}}(n)=in{\widehat {f}}(n)$.
If $f\in C^{k}(\mathbb {T} )$, then ${\widehat {f^{(k)}}}(n)=(in)^{k}{\widehat {f}}(n)$. In particular, since for a fixed $k\geq 1$ we have ${\widehat {f^{(k)}}}(n)\to 0$ as $n\to \infty$, it follows that $|n|^{k}{\widehat {f}}(n)$ tends to zero, which means that the Fourier coefficients converge to zero faster than the kth power of n for any $k\geq 1$.
If $c_{0},\,c_{\pm 1},\,c_{\pm 2},\ldots$ are coefficients and $\sum _{n=-\infty }^{\infty }|c_{n}|^{2}<\infty$ then there is a unique function $f\in L^{2}([-\pi ,\pi ])$ such that ${\hat {f}}(n)=c_{n}$ for every $n$.
Convolution theorems
The first convolution theorem states that if $f$ and $g$ are in $L^{1}([-\pi ,\pi ])$, the Fourier series coefficients of the 2?-periodic convolution of $f$ and $g$ are given by:
${\begin{aligned}\left[f*_{2\pi }g\right](x)\ &\triangleq \int _{-\pi }^{\pi }f(u)\cdot g[\operatorname {pv} (x-u)]\,du,&&{\big (}{\text{and }}\underbrace {\operatorname {pv} (x)\ \triangleq \operatorname {Arg} (e^{ix})} _{\text{principal value}}\,{\big )}\\&=\int _{-\pi }^{\pi }f(u)\cdot g(x-u)\,du,&&{\text{when }}g(x){\text{ is }}2\pi {\text{-periodic.}}\\&=\int _{2\pi }f(u)\cdot g(x-u)\,du,&&{\text{when both functions are }}2\pi {\text{-periodic, and the integral is over any }}2\pi {\text{ interval.}}\end{aligned}}$
The second convolution theorem states that the Fourier series coefficients of the product of $f$ and $g$ are given by the discrete convolution of the ${\hat {f}}$ and ${\hat {g}}$ sequences:
A doubly infinite sequence $\left\{c_{n}\right\}_{n\in Z}$ in $c_{0}(\mathbb {Z} )$ is the sequence of Fourier coefficients of a function in $L^{1}([0,2\pi ])$ if and only if it is a convolution of two sequences in $\ell ^{2}(\mathbb {Z} )$. See ^{[16]}
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 L^{2}(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 [-?,?] 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.
If the domain is not a group, then there is no intrinsically defined convolution. However, if $X$ is a compactRiemannian manifold, it has a Laplace-Beltrami operator. The Laplace-Beltrami operator is the differential operator that corresponds to Laplace operator for the Riemannian manifold $X$. Then, by analogy, one can consider heat equations on $X$. Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace-Beltrami operator as a basis. This generalizes Fourier series to spaces of the type $L^{2}(X)$, where $X$ is a Riemannian manifold. The Fourier series converges in ways similar to the $[-\pi ,\pi ]$ case. A typical example is to take $X$ to be the sphere with the usual metric, in which case the Fourier basis consists of spherical harmonics.
Locally compact Abelian groups
The generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups. However, there is a straightforward generalization to Locally Compact Abelian (LCA) groups.
This generalizes the Fourier transform to $L^{1}(G)$ or $L^{2}(G)$, where $G$ is an LCA group. If $G$ is compact, one also obtains a Fourier series, which converges similarly to the $[-\pi ,\pi ]$ case, but if $G$ is noncompact, one obtains instead a Fourier integral. This generalization yields the usual Fourier transform when the underlying locally compact Abelian group is $\mathbb {R}$.
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:
$f(x)$ designates a periodic function defined on $0<x\leq T$.
$a_{0},a_{n},b_{n}$ designate the Fourier Series coefficients (sine-cosine form) of the periodic function $f$ as defined in Eq.4.
An important question for the theory as well as applications is that of convergence. In particular, it is often necessary in applications to replace the infinite series $\sum _{-\infty }^{\infty }$ by a finite one,
$f_{N}(x)=\sum _{n=-N}^{N}{\hat {f}}(n)e^{inx}.$
This is called a partial sum. We would like to know, in which sense does $f_{N}(x)$ converge to $f(x)$ as $N\rightarrow \infty$.
Note that $f_{N}$ is a trigonometric polynomial of degree $N$. Parseval's theorem implies that
Theorem. The trigonometric polynomial $f_{N}$ is the unique best trigonometric polynomial of degree $N$ approximating $f(x)$, in the sense that, for any trigonometric polynomial $p\neq f_{N}$ of degree $N$, we have
Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.
Theorem. If $f$ belongs to $L^{2}(\left[-\pi ,\pi \right])$, then $f_{\infty }$ converges to $f$ in $L^{2}(\left[-\pi ,\pi \right])$, that is, $\|f_{N}-f\|_{2}$ converges to 0 as $N\rightarrow \infty$.
We have already mentioned that if $f$ is continuously differentiable, then $(i\cdot n){\hat {f}}(n)$ is the nth Fourier coefficient of the derivative $f'$. It follows, essentially from the Cauchy-Schwarz inequality, that $f_{\infty }$ is absolutely summable. The sum of this series is a continuous function, equal to $f$, since the Fourier series converges in the mean to $f$:
Theorem. If $f\in C^{1}(\mathbb {T} )$, then $f_{\infty }$ converges to $f$uniformly (and hence also pointwise.)
This result can be proven easily if $f$ is further assumed to be $C^{2}$, since in that case $n^{2}{\hat {f}}(n)$ tends to zero as $n\rightarrow \infty$. More generally, the Fourier series is absolutely summable, thus converges uniformly to $f$, provided that $f$ satisfies a Hölder condition of order $\alpha >1/2$. In the absolutely summable case, the inequality $\sup _{x}|f(x)-f_{N}(x)|\leq \sum _{|n|>N}|{\hat {f}}(n)|$ proves uniform convergence.
Many other results concerning the convergence of Fourier series are known, ranging from the moderately simple result that the series converges at $x$ if $f$ is differentiable at $x$, to Lennart Carleson's much more sophisticated result that the Fourier series of an $L^{2}$ 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".^{[18]}^{[19]}^{[20]}^{[21]}
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.^{[]} 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).
Laurent series - the substitution q = e^{ix} transforms a Fourier series into a Laurent series, or conversely. This is used in the q-series expansion of the j-invariant.
^Since the integral defining the Fourier transform of a periodic function is not convergent, it is necessary to view the periodic function and its transform as distributions. In this sense ${\mathcal {F}}\{e^{i{\frac {2\pi nx}{P}}}\}$ is a Dirac delta function, which is an example of a distribution.
^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.
^The scale factor is always equal to the period, 2? in this case.
^Fasshauer, Greg (2015). "Fourier Series and Boundary Value Problems"(PDF). Math 461 Course Notes, Ch 3. Department of Applied Mathematics, Illinois Institute of Technology. Retrieved 2020.
^Wilhelm Flügge, Stresses in Shells (1973) 2nd edition. ISBN978-3-642-88291-3. Originally published in German as Statik und Dynamik der Schalen (1937).
^Dorf, Richard C.; Tallarida, Ronald J. (1993). Pocket Book of Electrical Engineering Formulas (1st ed.). Boca Raton,FL: CRC Press. pp. 171-174. ISBN0849344735.
^Fourier, Jean-Baptiste-Joseph (1888). Gaston Darboux (ed.). Oeuvres de Fourier [The Works of Fourier] (in French). Paris: Gauthier-Villars et Fils. pp. 218-219 – via Gallica.
^ ^{a}^{b}^{c}^{d}^{e}Papula, Lothar (2009). Mathematische Formelsammlung: für Ingenieure und Naturwissenschaftler [Mathematical Functions for Engineers and Physicists] (in German). Vieweg+Teubner Verlag. ISBN978-3834807571.
William E. Boyce; Richard C. DiPrima (2005). Elementary Differential Equations and Boundary Value Problems (8th ed.). New Jersey: John Wiley & Sons, Inc. ISBN0-471-43338-1.
Joseph Fourier, translated by Alexander Freeman (2003). The Analytical Theory of Heat. Dover Publications. ISBN0-486-49531-0. 2003 unabridged republication of the 1878 English translation by Alexander Freeman of Fourier's work Théorie Analytique de la Chaleur, originally published in 1822.
Enrique A. Gonzalez-Velasco (1992). "Connections in Mathematical Analysis: The Case of Fourier Series". American Mathematical Monthly. 99 (5): 427-441. doi:10.2307/2325087. JSTOR2325087.
Katznelson, Yitzhak (1976). An introduction to harmonic analysis (Second corrected ed.). New York: Dover Publications, Inc. ISBN0-486-63331-4.
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.