Fourier series

 ( Joseph Fourier ) 

Independently of Fourier, astronomer Friedrich Wilhelm Bessel introduced Fourier series to solve Kepler's equation. His work was published in 1819, unaware of Fourier's work which remained unpublished until 1822.

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 eigenfunction. 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 and Bernhard Riemann

The harmonics are indexed by an integer, $n,$ which is also the number of cycles the corresponding sinusoids make in interval $P$. Therefore, the sinusoids have :

• a wavelength equal to $\backslash tfrac\{P\}\{n\}$ in the same units as $x$.
• a frequency equal to $\backslash tfrac\{n\}\{P\}$ in the reciprocal units of $x$.

These series can represent functions that are just a sum of one or more frequencies in the harmonic spectrum. In the limit $N\backslash to\backslash infty$, a trigonometric series can also represent the intermediate frequencies or non-sinusoidal functions because of the infinite number of terms.

The $\backslash tfrac\{2\}\{P\}$ scale factor follows from substituting into and utilizing the orthogonality of the trigonometric system. The equivalence of and follows from Euler's formula $$\backslash cos\; x\; =\; \backslash frac\{e^\{ix\}\; +\; e^\{-ix\}\}\{2\},\; \backslash quad\; \backslash sin\; x\; =\; \backslash frac\{e^\{ix\}\; -\; e^\{-ix\}\}\{2i\},$$ resulting in:

with $c\_\{0\}$ being the mean value of $s$ on the interval $P$. Conversely:

This example leads to a solution of the Basel problem.

where $A\_\{n\}$ is the amplitude and $\backslash varphi\_\{n\}$ is the phase shift of the $n^\{th\}$ harmonic.

The equivalence of and follows from the trigonometric identity: $$\backslash cos\backslash left(2\backslash pi\; \backslash tfrac\{n\}\{P\}x-\backslash varphi\_n\backslash right)\; =\; \backslash cos(\backslash varphi\_n)\backslash cos\backslash left(2\backslash pi\; \backslash tfrac\{n\}\{P\}\; x\backslash right)\; +\; \backslash sin(\backslash varphi\_n)\backslash sin\backslash left(2\backslash pi\; \backslash tfrac\{n\}\{P\}\; x\backslash right),$$ which implies

where $\backslash operatorname\{Arg\}(c\_n)$ is the argument of $c\_\{n\}$.

An example of determining the parameter $\backslash varphi\_n$ for one value of $n$ is shown in Figure 2. It is the value of $\backslash varphi$ at the maximum correlation between $s(x)$ and a cosine template, $\backslash cos(2\backslash pi\; \backslash tfrac\{n\}\{P\}\; x\; -\; \backslash varphi)$. The blue graph is the cross-correlation function, also known as a matched filter:

$\backslash begin\{align\}$

Fortunately, it is not necessary to evaluate this entire function, because its derivative is zero at the maximum: $$X\text{'}(\backslash varphi)\; =\; \backslash sin(\backslash varphi)\backslash cdot\; X(0)\; -\; \backslash cos(\backslash varphi)\backslash cdot\; X(\backslash pi/2)\; =\; 0,\; \backslash quad\; \backslash textrm\{at\}\backslash \; \backslash varphi\; =\; \backslash varphi\_n.$$ Hence $$\backslash varphi\_n\; \backslash equiv\; \backslash arctan(b\_n/a\_n)\; =\; \backslash arctan(X(\backslash pi/2)/X(0)).$$

$\backslash begin\{align\}$

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 Modulation a Dirac comb:

$S(f)\; \backslash \; \backslash triangleq\; \backslash \; \backslash sum\_\{n=-\backslash infty\}^\backslash infty\; Sn\backslash cdot\; \backslash delta\; \backslash left(f-\backslash frac\{n\}\{P\}\backslash right),$

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 $\backslash tfrac\{1\}\{P\}$, which is called the fundamental frequency. $s(x)$ can be recovered from this representation by an inverse Fourier transform:

$\backslash begin\{align\}$

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.

• $s(x)$ designates a periodic function with period $P.$
• $a\_0,\; a\_n,\; b\_n$ designate the Fourier series coefficients (sine-cosine form) of the periodic function $s(x).$

 \frac{-4A}{\pi}\frac{1}{n^2-1} & \quad n \text{ even} \\
 0 & \quad n \text{ odd}
 \end{cases}\\
     

 \frac{-2A}{\pi}\frac{1}{n^2-1} & \quad n \text{ even} \\
 0 & \quad n \text{ odd}
 \end{cases}\\
     

 \frac{A}{2} & \quad n=1 \\
 0 & \quad n > 1
 \end{cases}\\
     

• Complex conjugation is denoted by an asterisk.
• $s(x),r(x)$ designate $P$-periodic functions or functions defined only for $x\; \backslash in\; 0,P.$
• $Sn,\; Rn$ designate the Fourier series coefficients (exponential form) of $s$ and $r.$

From this, various relationships are apparent, for example :

• The transform of a real-valued function $(s\_\backslash mathrm\{RE\}+s\_\backslash mathrm\{RO\})$ is the conjugate symmetric function $S\_\backslash mathrm\{RE\}+i\backslash \; S\_\backslash mathrm\{IO\}.$ Conversely, a conjugate symmetric transform implies a real-valued time-domain.
• The transform of an imaginary-valued function $(i\backslash \; s\_\backslash mathrm\{IE\}+i\backslash \; s\_\backslash mathrm\{IO\})$ is the conjugate antisymmetric function $S\_\backslash mathrm\{RO\}+i\backslash \; S\_\backslash mathrm\{IE\},$ and the converse is true.
• The transform of a conjugate symmetric function $(s\_\backslash mathrm\{RE\}+i\backslash \; s\_\backslash mathrm\{IO\})$ is the real-valued function $S\_\backslash mathrm\{RE\}+S\_\backslash mathrm\{RO\},$ and the converse is true.
• The transform of a conjugate antisymmetric function $(s\_\backslash mathrm\{RO\}+i\backslash \; s\_\backslash mathrm\{IE\})$ is the imaginary-valued function $i\backslash \; S\_\backslash mathrm\{IE\}+i\backslash \; S\_\backslash mathrm\{IO\},$ and the converse is true.

• The pointwise product : $$h\_P(x)\; \backslash triangleq\; s\_P(x)\backslash cdot\; r\_P(x)$$ is also $P$-periodic, and its Fourier series coefficients are given by the discrete convolution of the $S$ and $R$ sequences : $$Hn\; =\; \backslash \{S*R\backslash \}n.$$
• The periodic convolution : $$h\_P(x)\; \backslash triangleq\; \backslash int\_\{P\}\; s\_P(\backslash tau)\backslash cdot\; r\_P(x-\backslash tau)\backslash ,\; d\backslash tau$$ is also $P$-periodic, with Fourier series coefficients : $$Hn\; =\; P\; \backslash cdot\; Sn\backslash cdot\; Rn.$$
• A doubly infinite sequence $\backslash left\; \backslash \{c\_n\; \backslash right\; \backslash \}\_\{n\; \backslash in\; Z\}$ in $c\_0(\backslash mathbb\{Z\})$ is the sequence of Fourier coefficients of a function in $L^1(0,2\backslash pi)$ if and only if it is a convolution of two sequences in $\backslash ell^2(\backslash mathbb\{Z\})$. See

• If $s\; \backslash in\; C^k(\backslash mathbb\{R\})$, then the Fourier coefficients of the $k^\{\backslash text\{th\}\}$ derivative of $s$ can be expressed in terms of the Fourier coefficients $\backslash widehat\{s\}n$ of $s$, via the formula $$\backslash widehat\{s^\{(k)\}\}n\; =\; (in)^k\; \backslash widehat\{s\}n.$$ In particular, since for any fixed $k\backslash geq\; 1$ we have $\backslash widehat\{s^\{(k)\}\}n\backslash to\; 0$ as $n\backslash to\backslash infty$, it follows that $|n|^k\backslash widehat\{s\}n$ tends to zero, i.e., the Fourier coefficients converge to zero faster than the $k^\{\backslash text\{th\}\}$ power of $|n|$.

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.

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 $-\backslash pi,\backslash pi$ 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 $\backslash mathbb\{R\}$.

This follows from an earlier and more concrete representation of a Radon measure (i.e. a locally finite Borel measure) on $\backslash mathbb\{R\}$, given by F. Riesz. That is, if $F$ is function of bounded variation on the interval $0,P$ then the Fourier coefficients can be expressed by the Riemann-Stieltjes integral $$c\_n\; =\; \backslash frac\{1\}\{P\}\backslash int\_0^P\; \backslash \; e^\{-i\; 2\backslash pi\; \backslash tfrac\{n\}\{P\}\; x\; \}\backslash ,dF(x),\; \backslash quad\; \backslash forall\; n\backslash in\backslash mathbb\{Z\},$$ called the Fourier-Stieltjes coefficients of $F$. As the distributional derivative of $F$ is a Radon measure, it is subject to the Lebesgue decomposition and can be expressed as $dF\; =\; F\text{'}dx\; +\; dF\_s$. If $dF\_s\; =\; 0$ the expression reduces to the original definition of the Fourier coefficients, hence a Fourier series is a Fourier-Stieltjes series.

The question whether or not $\backslash mu$ exists for a given sequence of $c\_n$ forms the basis of the trigonometric moment problem.

The Fourier series can be generalized still further from measures to distributions. If the Fourier coefficients are determined by a distribution $F\; \backslash in\; \backslash mathcal\{D\}\text{'}$ then the series is sometimes described as a Fourier-Schwartz series.

While it is often extremely difficult to decide whether a given series is a Fourier or a Fourier-Stieltjes series, deciding whether or not it is a Fourier-Schwartz series is relatively trivial.

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, a discrete form of the Fourier cosine transform, which uses only cosine as the basis function.

For two-dimensional arrays with a staggered appearance, half of the Fourier series coefficients disappear, due to additional symmetry. Vanishing of Half the Fourier Coefficients in Staggered Arrays

In order to develop $f(\backslash mathbf\{r\})$ in a Fourier series, it is convenient to introduce an auxiliary function $$g(x\_1,x\_2,x\_3)\; \backslash triangleq\; f(\backslash mathbf\{r\})\; =\; f\; \backslash left\; (x\_1\backslash frac\{\backslash mathbf\{a\}\_\{1\}\}\{a\_1\}+x\_2\backslash frac\{\backslash mathbf\{a\}\_\{2\}\}\{a\_2\}+x\_3\backslash frac\{\backslash mathbf\{a\}\_\{3\}\}\{a\_3\}\; \backslash right\; ).$$ Both $f(\backslash mathbf\{r\})$ and $g(x\_1,x\_2,x\_3)$ contain essentially the same information. However, instead of the position vector $\backslash mathbf\{r\}$, the arguments of $g$ are coordinates $x\_\{1,2,3\}$ along the unit vectors $\backslash mathbf\{a\}\_\{i\}/\{a\_i\}$ of the Bravais lattice, such that $g$ is an ordinary periodic function in these variables,$$g(x\_1,x\_2,x\_3)\; =\; g(x\_1+a\_1,x\_2,x\_3)\; =\; g(x\_1,x\_2+a\_2,x\_3)\; =\; g(x\_1,x\_2,x\_3+a\_3)\backslash quad\backslash forall\backslash ;x\_1,x\_2,x\_3.$$ This trick allows us to develop $g$ as a multi-dimensional Fourier series, in complete analogy with the square-periodic function discussed in the previous section. Its Fourier coefficients are$$\backslash begin\{align\}\; c(m\_1,\; m\_2,\; m\_3)$$

= \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)\, e^{-i 2\pi \left(\tfrac{m_1}{a_1} x_1+\tfrac{m_2}{a_2} x_2 + \tfrac{m_3}{a_3} x_3\right)}
     

Once we have these coefficients, the function $g$ can be recovered via the Fourier series $$g(x\_1,\; x\_2,\; x\_3)=\backslash sum\_\{m\_1,\; m\_2,\; m\_3\; \backslash in\; \backslash Z\; \}\; \backslash ,c(m\_1,\; m\_2,\; m\_3)\; \backslash ,\; e^\{i\; 2\backslash pi\; \backslash left(\; \backslash tfrac\{m\_1\}\{a\_1\}\; x\_1+\; \backslash tfrac\{m\_2\}\{a\_2\}\; x\_2\; +\; \backslash tfrac\{m\_3\}\{a\_3\}\; x\_3\backslash right)\}.$$ We would now like to abandon the auxiliary coordinates $x\_\{1,2,3\}$ and to return to the original position vector $\backslash mathbf\{r\}$. This can be achieved by means of the reciprocal lattice whose vectors $\backslash mathbf\{b\}\_\{1,2,3\}$ are defined such that they are orthonormal (up to a factor $2\backslash pi$) to the original Bravais vectors $\backslash mathbf\{a\}\_\{1,2,3\}$, $$\backslash mathbf\{a\}\_i\backslash cdot\backslash mathbf\{b\_j\}=2\backslash pi\backslash delta\_\{ij\},$$with $\backslash delta\_\{ij\}$ the Kronecker delta. With this, the scalar product between a reciprocal lattice vector $\backslash mathbf\{Q\}$ and an arbitrary position vector $\backslash mathbf\{r\}$ written in the Bravais lattice basis becomes $$\backslash mathbf\{Q\}\; \backslash cdot\; \backslash mathbf\{r\}\; =\; \backslash left\; (\; m\_1\backslash mathbf\{b\}\_1\; +\; m\_2\backslash mathbf\{b\}\_2\; +\; m\_3\backslash mathbf\{b\}\_3\; \backslash right\; )\; \backslash cdot\; \backslash left\; (x\_1\backslash frac\{\backslash mathbf\{a\}\_1\}\{a\_1\}+\; x\_2\backslash frac\{\backslash mathbf\{a\}\_2\}\{a\_2\}\; +x\_3\backslash frac\{\backslash mathbf\{a\}\_3\}\{a\_3\}\; \backslash right\; )\; =\; 2\backslash pi\; \backslash left(\; x\_1\backslash frac\{m\_1\}\{a\_1\}+x\_2\backslash frac\{m\_2\}\{a\_2\}+x\_3\backslash frac\{m\_3\}\{a\_3\}\; \backslash right\; ),$$which is exactly the expression occurring in the Fourier exponents. The Fourier series for $f(\backslash mathbf\{r\})\; =g(x\_1,x\_2,x\_3)$ can therefore be rewritten as a sum over the all reciprocal lattice vectors $\backslash mathbf\{Q\}=\; m\_1\backslash mathbf\{b\}\_1+m\_2\backslash mathbf\{b\}\_2+m\_3\backslash mathbf\{b\}\_3$,$$f(\backslash mathbf\{r\})=\backslash sum\_\{\backslash mathbf\{Q\}\}\; c(\backslash mathbf\{Q\})\backslash ,\; e^\{i\; \backslash mathbf\{Q\}\; \backslash cdot\; \backslash mathbf\{r\}\},$$ and the coefficients are$$c(\backslash mathbf\{Q\})\; =\; \backslash frac\{1\}\{a\_3\}\; \backslash int\_0^\{a\_3\}\; dx\_3\; \backslash ,\; \backslash frac\{1\}\{a\_2\}\backslash int\_0^\{a\_2\}\; dx\_2\; \backslash ,\; \backslash frac\{1\}\{a\_1\}\backslash int\_0^\{a\_1\}\; dx\_1\; \backslash ,\; f\backslash left(x\_1\backslash frac\{\backslash mathbf\{a\}\_1\}\{a\_1\}\; +\; x\_2\backslash frac\{\backslash mathbf\{a\}\_2\}\{a\_2\}\; +\; x\_3\backslash frac\{\backslash mathbf\{a\}\_3\}\{a\_3\}\; \backslash right)\; e^\{-i\; \backslash mathbf\{Q\}\; \backslash cdot\; \backslash mathbf\{r\}\}.$$ The remaining task will be to convert this integral over lattice coordinates back into a volume integral. The relation between the lattice coordinates $x\_\{1,2,3\}$ and the original cartesian coordinates $\backslash mathbf\{r\}\; =\; (x,y,z)$ is a linear system of equations, $$\backslash mathbf\{r\}\; =\; x\_1\backslash frac\{\backslash mathbf\{a\}\_1\}\{a\_1\}+x\_2\backslash frac\{\backslash mathbf\{a\}\_2\}\{a\_2\}+x\_3\backslash frac\{\backslash mathbf\{a\}\_3\}\{a\_3\},$$which, when written in matrix form, $$\backslash begin\{bmatrix\}x\backslash \backslash y\backslash \backslash z\backslash end\{bmatrix\}\; =\backslash mathbf\{J\}\backslash begin\{bmatrix\}x\_1\backslash \backslash x\_2\backslash \backslash x\_3\backslash end\{bmatrix\}\; =\backslash begin\{bmatrix\}\backslash frac\{\backslash mathbf\{a\}\_1\}\{a\_1\},\backslash frac\{\backslash mathbf\{a\}\_2\}\{a\_2\},\backslash frac\{\backslash mathbf\{a\}\_3\}\{a\_3\}\backslash end\{bmatrix\}\backslash begin\{bmatrix\}x\_1\backslash \backslash x\_2\backslash \backslash x\_3\backslash end\{bmatrix\}\backslash ,,$$involves a constant matrix $\backslash mathbf\{J\}$ whose columns are the unit vectors $\backslash mathbf\{a\}\_j/a\_j$ of the Bravais lattice. When changing variables from $\backslash mathbf\{r\}$ to $(x\_1,x\_2,x\_3)$ in an integral, the same matrix $\backslash mathbf\{J\}$ appears as a Jacobian matrix

Its determinant $J$ is therefore also constant and can be inferred from any integral over any domain; here we choose to calculate the volume of the primitive unit cell $\backslash Gamma$ in both coordinate systems: $$V\_\{\backslash Gamma\}\; =\; \backslash int\_\{\backslash Gamma\}\; d^3\; r\; =\; J\; \backslash int\_\{0\}^\{a\_1\}\; dx\_1\; \backslash int\_\{0\}^\{a\_2\}\; dx\_2\; \backslash int\_\{0\}^\{a\_3\}\; dx\_3=J\backslash ,\; a\_1\; a\_2\; a\_3$$ The unit cell being a parallelepiped, we have $V\_\{\backslash Gamma\}=\backslash mathbf\{a\}\_1\backslash cdot(\backslash mathbf\{a\}\_2\; \backslash times\; \backslash mathbf\{a\}\_3)$ and thus $$d^3r=J\; dx\_1\; dx\_2\; dx\_3\; =\backslash frac\{\backslash mathbf\{a\}\_1\backslash cdot(\backslash mathbf\{a\}\_2\; \backslash times\; \backslash mathbf\{a\}\_3)\}\{a\_1\; a\_2\; a\_3\}\; dx\_1\; dx\_2\; dx\_3.$$ This allows us to write $c\; (\backslash mathbf\{Q\})$ as the desired volume integral over the primitive unit cell $\backslash Gamma$ in ordinary cartesian coordinates: $$c(\backslash mathbf\{Q\})\; =\; \backslash frac\{1\}\{\backslash mathbf\{a\}\_1\backslash cdot(\backslash mathbf\{a\}\_2\; \backslash times\; \backslash mathbf\{a\}\_3)\}\backslash int\_\{\backslash Gamma\}\; d^3\; r\backslash ,\; f(\backslash mathbf\{r\})\backslash cdot\; e^\{-i\; \backslash mathbf\{Q\}\; \backslash cdot\; \backslash mathbf\{r\}\}\backslash ,.$$

Fourier_series_square_wave_circles_animation.gif|link=//upload.wikimedia.org/wikipedia/commons/b/bd/Fourier_series_square_wave_circles_animation.svg|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 Fourier_series_sawtooth_wave_circles_animation.gif|link=//upload.wikimedia.org/wikipedia/commons/1/1e/Fourier_series_sawtooth_wave_circles_animation.svg|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 Example_of_Fourier_Convergence.gif |Example of convergence to a somewhat arbitrary function. Note the development of the "ringing" (Gibbs phenomenon) at the transitions to/from the vertical sections.

The theorems proving that a Fourier series is a valid representation of any periodic function (that satisfies the Dirichlet conditions), and informal variations of them that do not specify the convergence conditions, are sometimes referred to generically as Fourier's theorem or the Fourier theorem.