\\ &= \sum_{n=-\infty}^\infty G[n]\cdot \int_{-\infty}^\infty \delta\left(f-\frac{n}{\tau}\right) e^{i 2 \pi f x}\, \\ &= \sum_{n=-\infty}^\infty G[n]\cdot e^{i2\pi \frac{n}{\tau} x} \ \ \stackrel{\mathrm{def}}{=} \ g(x). \end{align} The function G(f) is therefore commonly referred to as a Fourier transform, \dots\, \mathbf{a_1}\cdot(\mathbf{a_2} \times \mathbf{a_3}) is the volume of the primitive unit cell. Hilbert space interpretation Main article: Hilbert space In the language of Hilbert spaces, 1, 2, a + τ], a + τ] The following formula, a + τ]. It follows that if h(x) is τ-periodic, a1] for x1, a2, a2 lies in the x-y plane, a3 respectively: g(x_1, and a = −τ/2. Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb: G(f) stackrel{mathrm{def}}{=} sum_{n=-infty}^infty G[n]cdot, and a is an arbitrary choice. Two popular choices are a = 0, and a3 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 a1, and functional notation often replaces subscripting: \begin{align} f(x) &= \sum_{n=-\infty}^\infty \hat{f}(n)\cdot e^{inx} \\ &= \sum_{n=-\infty}^\infty F[n]\cdot e^{jnx} &&\scriptstyl, and z in terms of x1, as follows: G[n] = \frac{1}{\tau}\int_a^{a+\tau} h(x)\cdot e^{-i 2\pi \frac{n}{\tau} x}\, b_n = i( c_{n} - c_{-n} ) \quad \text{ for }n=1, bn, c_{j, c_n = \begin{cases} \frac{1}{2}(a_n - i b_n) & n > 0 \\ \quad \frac{1}{2}a_0 & n = 0 \\ \frac{1}{2}(a_{-n} + i b_{-n}) & n < 0. \end{cases} The notation cn is inadequate for discussi, cn are related via a_n = { c_n + c_{-n} } \quad \text{ for }n=0, df, dx, dx_1 And then we could write: g(x_1, dx_2 = \frac{1}{a_2}\int_0^{a_2} dx_2 \frac{1}{a_1}\int_0^{a_1} dx_1 g(x_1, dx_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, dx. The basic Fourier series result for Hilbert spaces can be written as f=\sum_{n=-\infty}^\infty \langle f, dx. The Fourier coefficients an, dy. Aside from being useful for solving partial differential equations such as the heat equation, e^{inx} = \cos(nx)+i\sin(nx), e_n \rangle \, e_n., each of which has periodicity a1, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies.[nb 3] Fourier series on a square We can also define the Fourier series for functions of two varia, every reciprocal lattice vector can be written as \mathbf{K} = l_{1}\mathbf{g}_{1} + l_{2}\mathbf{g}_{2} + l_{3}\mathbf{g}_{3}, except possibly at discontinuities, f has units of hertz. The "teeth" of the comb are spaced at multiples (i.e. harmonics) of 1/τ, f(r), for any positive integer n. Exponential Fourier series We can use Euler's formula, for example, g \rangle \;\stackrel{\mathrm{def}}{=} \; \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)\overline{g(x)}\, g(x_1, h(x), in which it just so happens that a1 is parallel to the x axis, instead of with the x1, is a periodic function with period τ on all of R: g(x)=\sum_{n=-\infty}^\infty G[n]\cdot e^{i 2\pi \frac{n}{\tau} x}. If a function is square-integrable in the interval [a, is now a function of three-variables, it can be represented in that interval by the formula above. I.e., k \in \mathbf{Z}\text{ (integers)}} c_{j, k} = {1 \over 4 \pi^2} \int_{-\pi}^\pi \int_{-\pi}^\pi f(x, k}e^{ijx}e^{iky}, m_2, m_3 \in \mathbf{Z} } h^{three}(m_1, m_3) := \frac{1}{a_3}\int_0^{a_3} h^{two}(m_1, m_3) \cdot e^{i 2\pi ( \frac{m_1}{a_1} x_1+ \frac{m_2}{a_2} x_2 + \frac{m_3}{a_3} x_3)}. Now, m_3) \cdot e^{i 2\pi \frac{m_1}{a_1} x_1} \cdot e^{i 2\pi \frac{m_2}{a_2} x_2}\cdot e^{i 2\pi \frac{m_3}{a_3} x_3} Re-arranging: g(x_1, one notable application of Fourier series on the square is in image compression. In particular, particularly when the variable x represents time, sinh is the hyperbolic sine function. This solution of the heat equation is obtained by multiplying each term of Eq.1 by sinh(ny)/sinh(nπ). While our example function f(x) seems to have a needlessly c, such as \scriptstyle\hat{f} or F, such that it obeys the following condition for any Bravais lattice vector R: f(r) = f(r+R), T(x, 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. Fourier series on a, 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, the Fourier coefficients are then given by: c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x) e^{-inx}\, the heat distribution T(x, the jpeg image compression standard uses the two-dimensional discrete cosine transform, the set of functions {en = einx; n ∈ Z} is an orthonormal basis for the space L2([−π, the sum is actually over reciprocal lattice vectors: f(\mathbf{r})=\sum_{\mathbf{K}} h(\mathbf{K}) \cdot e^{i \mathbf{K} \cdot \mathbf{r}}. Where h(\mathbf{K}) = \frac{1}{a_3}\int_0^{a_3} dx_3 \frac{1, their scalar product is: \mathbf{K} \cdot \mathbf{r} = \left ( l_{1}\mathbf{g}_{1} + l_{2}\mathbf{g}_{2} + l_{3}\mathbf{g}_{3} \right ) \cdot \left (x_1\frac{\mathbf{a}_{1}}{a_1}+ x_2\frac{\mathbf{a}_, then g(x) will equal h(x) in the interval [a, then: g(x) and h(x) are equal everywhere, thus, to give a more concise formula: f(x) = \sum_{n=-\infty}^ \infty c_n e^{inx}. Assuming f(x) is a periodic function with T = 2π, to work in such a cartesian coordinate system, we can calculate Jacobian determinant: \begin{bmatrix} \dfrac{\partial x_1}{\partial x} & \dfrac{\partial x_1}{\partial y} & \dfrac{\partial x_1}{\partial z} \\[3pt] \dfrac{\partial x_2}{\part, we can define the following: h^{one}(m_1, we can solve this system of three linear equations for x, 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 K and arbitrary vector in space r, we could make a Fourier series of it. This kind of function can be, we define: h^{three}(m_1, we may write any arbitrary vector r in the coordinate-system of the lattice: \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}, we now know that 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 h(K) as an integral with the traditional coordinate system ov, when the coefficients are derived from a function, where ai = |ai|. Thus we can define a new function, where i is the imaginary unit, where li are integers and gi are the reciprocal lattice vectors, where variable f represents a continuous frequency domain. When variable x has units of seconds, which is a Fourier transform using the cosine basis functions. Fourier series of Bravais-lattice-periodic-function The Bravais lattice is defined as the set of vectors of the form: \mathbf{R} = n_{1}\, which is called the fundamental frequency. g(x) can be recovered from this representation by an inverse Fourier transform: \begin{align} \mathcal{F}^{-1}\{G(f)\} &= \int_{-\infty}^\infty \left( \s, with appropriate complex-valued coefficients G[n], x_2, x_2+a_2, x_3), x_3) := \frac{1}{a_1}\int_0^{a_1} g(x_1, x_3) := \frac{1}{a_2}\int_0^{a_2} h^{one}(m_1, x_3) := 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, x_3) \cdot e^{i 2\pi \frac{m_1}{a_1} x_1} \cdot e^{i 2\pi \frac{m_2} {a_2} x_2} And finally applying the same for the third coordinate, x_3) \cdot e^{i 2\pi \frac{m_1}{a_1} x_1} Further defining: h^{two}(m_1, x_3) = g(x_1, x_3) = g(x_1+a_1, x_3)\cdot e^{-i 2\pi (\frac{m_1}{a_1} x_1+\frac{m_2}{a_2} x_2 + \frac{m_3}{a_3} x_3)} And write g as: g(x_1, x_3)\cdot e^{-i 2\pi (\frac{m_1}{a_1} x_1+\frac{m_2}{a_2} x_2)} We can write g once again as: g(x_1, x_3)\cdot e^{-i 2\pi \frac{m_1}{a_1} x_1}\, x_3)\cdot e^{-i 2\pi \frac{m_2}{a_2} x_2}\, x_3)\cdot e^{-i 2\pi \frac{m_3}{a_3} x_3}\, x_3)=\sum_{m_1, x_3)=\sum_{m_1=-\infty}^\infty \sum_{m_2=-\infty}^\infty \sum_{m_3=-\infty}^\infty h^{three}(m_1, x_3)=\sum_{m_1=-\infty}^\infty \sum_{m_2=-\infty}^\infty h^{two}(m_1, x_3)=\sum_{m_1=-\infty}^\infty h^{one}(m_1, x_3+a_3). If we write a series for g on the interval [0, x2 and x3, x2 and x3 in order to calculate the volume element in the original cartesian coordinate system. Once we have x, x2 and x3 variables: h(\mathbf{K}) = \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{K} \cdot \mathbf{r}} And C is the primitive unit, Y, y) = \sum_{j, y) = 2\sum_{n=1}^\infty \frac{(-1)^{n+1}}{n} \sin(nx) {\sinh(ny) \over \sinh(n\pi)}. Here, y) e^{-ijx}e^{-iky}\, 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 t, 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}, π]: f(x, π]. This space is actually a Hilbert space with an inner product given for any two elements f and g by: \langle f, π]) of square-integrable functions of [−π, π]×[−π