Fourier series
Fourier series
| Fourier transforms |
|---|
| Continuous Fourier transform |
| Fourier series |
| Discrete-time Fourier transform |
| Discrete Fourier transform |
| Discrete Fourier transform over a ring |
| Fourier analysis |
| Related transforms |
In mathematics, a Fourier series (/ˈfʊrieɪ, -iər/)[5] 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 the 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.
| Fourier transforms |
|---|
| Continuous Fourier transform |
| Fourier series |
| Discrete-time Fourier transform |
| Discrete Fourier transform |
| Discrete Fourier transform over a ring |
| Fourier analysis |
| Related transforms |
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.[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 (continuous)[6] 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.[7] 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[8] and Bernhard Riemann[9][10][11] 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,[12] thin-walled shell theory,[13] etc.
Definition
If is a function contained in an interval of length (and zero elsewhere), the upper-right quadrant is an example of what its Fourier series coefficients () might look like when plotted against their corresponding harmonic frequencies. The upper-left quadrant is the corresponding Fourier transform of The Fourier series summation (not shown) synthesizes a periodic summation of whereas the inverse Fourier transform (not shown) synthesizes only
- andand
![{\displaystyle x\in [0,1],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c68354ed86bae40d711eba3ef26c4ec740fcc8fc)
![{\displaystyle x\in [-\pi ,\pi ],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f68d2390bdd36e72e19b37682380f02214d52b2)
- Ifis-periodic, then any interval of that length is sufficient.
- andcan be reduced toand.
- Many texts chooseto simplify the argument of the sinusoid functions.
The synthesis process (the actual Fourier series) is:
Using a trigonometric identity:
Therefore, with definitions:
the final result is:
Complex-valued functions
- and
![{\displaystyle {\begin{aligned}c_{n}&={\frac {1}{P}}\int _{P}\operatorname {Re} \{s(x)\}\cdot e^{-i{\tfrac {2\pi nx}{P}}}\ dx+i\cdot {\frac {1}{P}}\int _{P}\operatorname {Im} \{s(x)\}\cdot e^{-i{\tfrac {2\pi nx}{P}}}\ 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 nx}{P}}}\ dx\ =\ {\frac {1}{P}}\int _{P}s(x)\cdot e^{-i{\tfrac {2\pi nx}{P}}}\ dx.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32ab5a1982824b357f94d5fe7410bfb84fef8e6b)
Other common notations
![{\displaystyle S[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b4222b53917b43f530116997b71049100c95586)
![{\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^{j\,2\pi nx/P}&&\scriptstyle {\mathsf {common\ engineering\ notation}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a0ce35db68aa1a4984b746f18928b5d8b6eebfc)
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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6a4127cd2d8d239076aad35e6b82248554b6036)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/446316db0d563e8ceac0da5d5d078b67f0c0f4e5)
Convergence
![{\displaystyle [x_{0},x_{0}+P]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef79a1b836ec65eacb0d2c73464996d2b7830ba)
Examples
Example 1: a simple Fourier series
Plot of the sawtooth wave, a periodic continuation of the linear function on the interval
Animated plot of the first five successive partial Fourier series
Heat distribution in a metal plate, using Fourier's method
We now use the formula above to give a Fourier series expansion of a very simple function. Consider a sawtooth wave
In this case, the Fourier coefficients are given by
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cd645b661c531998472837b4709b23bf2d2c939)
| **(Eq.7)** |
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c051045100b8d31eada1c7fcfd414759404ed25b)
This example leads us to a solution to the Basel problem.
Example 2: Fourier's motivation
![{\displaystyle (x,y)\in [0,\pi ]\times [0,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c18734f151b17b5d3e325f79c7000826ab832610)
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
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
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
![{\displaystyle [-\pi ,\pi ]\times [-\pi ,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df436805f50de7386abdb2a9d058672ec1b4cebb)
![{\displaystyle {\begin{aligned}f(x,y)&=\sum _{j,k\,\in \,\mathbb {Z} {\text{ (integers)}}}c_{j,k}e^{ijx}e^{iky},\\[5pt]c_{j,k}&={1 \over 4\pi ^{2}}\int _{-\pi }^{\pi }\int _{-\pi }^{\pi }f(x,y)e^{-ijx}e^{-iky}\,dx\,dy.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71f5aaa23f00684f16ecbf31838ff5ac61f807db)
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:
Thus we can define a new function,
If we write a series for g on the interval [0, a1] for x1, we can define the following:
And then we can write:
Further defining:
![\begin{align}
h^\mathrm{two}(m_1, m_2, x_3) & := \frac{1}{a_2}\int_0^{a_2} h^\mathrm{one}(m_1, x_2, x_3)\cdot e^{-i 2\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^{-i 2\pi \left(\frac{m_1}{a_1} x_1+\frac{m_2}{a_2} x_2\right)}
\end{align}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b375ad1bad97e4cee6cfa9c82d20582af5ba6b4)
Finally applying the same for the third coordinate, we define:
![\begin{align}
h^\mathrm{three}(m_1, m_2, m_3) & := \frac{1}{a_3}\int_0^{a_3} h^\mathrm{two}(m_1, m_2, x_3)\cdot e^{-i 2\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^{-i 2\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{align}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bac3a4c30ff78f3abc70b6b65a6274b8899ca038)
Re-arranging:
And so it is clear that in our expansion, the sum is actually over reciprocal lattice vectors:
where
Assuming
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e5df9134486606d6a55c8ec4a96ee3ca353e924)
which after some calculation and applying some non-trivial cross-product identities can be shown to be equal to:
Hilbert space interpretation
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 , or the functions are different, and pi only if and are equal, and the function used is the same.
![{\displaystyle L^{2}([-\pi ,\pi ])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f84fea7a212acaf14649b6cdcca282b0646a8b0)
![[-\pi ,\pi ]](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb064fd6c55820cfa660eabeeda0f6e3c4935ae6)
The basic Fourier series result for Hilbert spaces can be written as
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:
(where δmn is the Kronecker delta), and
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:
is the complex conjugate of .
designate -periodic functions defined on .
designate the Fourier series coefficients (exponential form) of and as defined in equation Eq.5.
| Property | Time domain | Frequency domain (exponential form) | Remarks | Reference |
|---|---|---|---|---|
| Linearity | complex numbers | |||
| Time reversal / Frequency reversal | [17]:p. 610 | |||
| Time conjugation | [17]:p. 610 | |||
| Time reversal & conjugation | ||||
| Real part in time | ||||
| Imaginary part in time | ||||
| Real part in frequency | ||||
| Imaginary part in frequency | ||||
| Shift in time / Modulation in frequency | real number | [17]:p. 610 | ||
| Shift in frequency / Modulation in time | integer | [17]:p. 610 |
![{\displaystyle a\cdot F[n]+b\cdot G[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc955a6cf808d4921d875ff75eb9a0ef841fecc9)
![{\displaystyle F[-n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/33b4a4823263d9c8be71e81b434efa1e0e7f7b78)
![{\displaystyle F[-n]^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a0bb8e1ac85b41420733bbc3176752128cb2ad1)
![{\displaystyle F[n]^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a49c30283e8c78c1a9bedb82bc10d910060b88d)
![{\displaystyle {\frac {1}{2}}(F[n]+F[-n]^{*})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/561b550cd40e3a9b9094047184114aa358f1a96b)
![{\displaystyle {\frac {1}{2i}}(F[n]-F[-n]^{*})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2317f4fc3c68ae6247b484faefb4afcfb05dfbb)
![{\displaystyle \operatorname {Re} {(F[n])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46139e55f1d3fefaedc2c50c143c2070a71cdb80)
![{\displaystyle \operatorname {Im} {(F[n])}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b17cae376b2d16a948caa8532ba478bc7b78dbf)
![{\displaystyle F[n]\cdot e^{-i{\frac {2\pi x_{0}}{T}}n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0867072969dbc9aa6f27cca71a9ffeb6a6853a75)
![{\displaystyle F[n-n_{0}]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01ff33ce94b62cdac1424cda391a7a63d678bb78)
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:[18]
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 (i fIE+ i fIO) is the odd symmetric function FRO+ i FIE, and the converse is true.
The transform of an even-symmetric function (fRE+ i fIO) is the real-valued function FRE+ FRO, and the converse is true.
The transform of an odd-symmetric function (fRO+ i fIE) is the imaginary-valued function i FIE+ i FIO, and the converse is true.
Riemann–Lebesgue lemma
Derivative property
If , then the Fourier coefficients of the derivative can be expressed in terms of the Fourier coefficients of the function , via the formula .
If , then . In particular, since tends to zero, we have that tends to zero, which means that the Fourier coefficients converge to zero faster than the kth power of n.
Parseval's theorem
Plancherel's theorem
![f\in L^2([-\pi,\pi])](https://wikimedia.org/api/rest_v1/media/math/render/svg/74d27747bf5e28fa54e94f218bd9296213ef25c5)
Convolution theorems
The first convolution theorem states that if and are in , the Fourier series coefficients of the 2π-periodic convolution of and are given by:
 = 2\pi\cdot \hat{f}(n)\cdot\hat{g}(n),](https://wikimedia.org/api/rest_v1/media/math/render/svg/956ea404aa998519c088c08c9a9e9f29c9bd38f0)
- where:
\ &\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2105aae4f5eb50bf6d6541e8a28ef6d0a75db5)
The second convolution theorem states that the Fourier series coefficients of the product of and are given by the discrete convolution of the and sequences:
 = [\hat{f}*\hat{g}](n).](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a01ca431cbe51c2353a325265bb70204e5a1ae1)
A doubly infinite sequence in is the sequence of Fourier coefficients of a function in if and only if it is a convolution of two sequences in . See [19]
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
The atomic orbitals of chemistry are partially described by spherical harmonics, which can be used to produce Fourier series on the sphere.
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.
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:
designates a periodic function defined on .
designate the Fourier Series coefficients (sine-cosine form) of the periodic function as defined in Eq.4.
Approximation and convergence of Fourier series
Least squares property
Theorem. The trigonometric polynomial is the unique best trigonometric polynomial of degree approximating , in the sense that, for any trigonometric polynomial of degree , we have
where the Hilbert space norm is defined as:
Convergence
Because of the least squares property, and because of the completeness of the Fourier basis, we obtain an elementary convergence result.
![{\displaystyle L^{2}(\left[-\pi ,\pi \right])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e9f22db3f847b6cab3b1fb2f43e32dd574acc1f)
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).
See also
ATS theorem
Dirichlet kernel
Discrete Fourier transform
Fejér's theorem
Fourier analysis
Fourier sine and cosine series
Gibbs phenomenon
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.
Multidimensional transform
Spectral theory
Sturm–Liouville theory