The delta function can be viewed as the limit of a sequence of functions \delta (x) = \lim_{\varepsilon\to 0^+} \eta_\varepsilon(x). This limit is meant in a weak sense: either that {{NumBlk2|:| \lim_{\varepsilon\to 0^+} \int_{-\infty}^\infty \eta_\varepsilon(x)f(x) \, dx = f(0) |5}} for all
continuous functions having
compact support, or that this limit holds for all
smooth functions with compact support. The former is convergence in the
vague topology of measures, and the latter is convergence in the sense of
distributions.
Approximations to the identity An approximate delta function can be constructed in the following manner. Let be an absolutely integrable function on of total integral , and define \eta_\varepsilon(x) = \varepsilon^{-1} \eta \left (\frac{x}{\varepsilon} \right). In dimensions, one uses instead the scaling \eta_\varepsilon(x) = \varepsilon^{-n} \eta \left (\frac{x}{\varepsilon} \right). Then a simple change of variables shows that also has integral . One may show that () holds for all continuous compactly supported functions , and so converges weakly to in the sense of measures. The constructed in this way are known as an
approximation to the identity. This terminology is because the space of absolutely integrable functions is closed under the operation of
convolution of functions: whenever and are in . However, there is no identity in for the convolution product: no element such that for all . Nevertheless, the sequence does approximate such an identity in the sense that f*\eta_\varepsilon \to f \quad \text{as }\varepsilon\to 0. This limit holds in the sense of
mean convergence (convergence in ). Further conditions on the , for instance that it be a mollifier associated to a compactly supported function, are needed to ensure pointwise convergence
almost everywhere. If the initial is itself smooth and compactly supported then the sequence is called a
mollifier. The standard mollifier is obtained by choosing to be a suitably normalized
bump function, for instance \eta(x) = \begin{cases} \frac{1}{I_n} \exp\Big( -\frac{1}{1-|x|^2} \Big) & \text{if } |x| (I_n ensuring that the total integral is 1). In some situations such as
numerical analysis, a
piecewise linear approximation to the identity is desirable. This can be obtained by taking to be a
hat function. With this choice of , one has \eta_\varepsilon(x) = \varepsilon^{-1}\max \left (1-\left|\frac{x}{\varepsilon}\right|,0 \right) which are all continuous and compactly supported, although not smooth and so not a mollifier.
Probabilistic considerations In the context of
probability theory, it is natural to impose the additional condition that the initial in an approximation to the identity should be positive, as such a function then represents a
probability distribution. Convolution with a probability distribution is sometimes favorable because it does not result in
overshoot or undershoot, as the output is a
convex combination of the input values, and thus falls between the maximum and minimum of the input function. Taking to be any probability distribution at all, and letting as above will give rise to an approximation to the identity. In general this converges more rapidly to a delta function if, in addition, has mean and has small higher moments. For instance, if is the
uniform distribution on {{nowrap|1=\left[-\frac{1}{2},\frac{1}{2}\right],}} also known as the
rectangular function, then: \eta_\varepsilon(x) = \frac{1}{\varepsilon}\operatorname{rect}\left(\frac{x}{\varepsilon}\right)= \begin{cases} \frac{1}{\varepsilon},&-\frac{\varepsilon}{2} Another example is with the
Wigner semicircle distribution \eta_\varepsilon(x)= \begin{cases} \frac{2}{\pi \varepsilon^2}\sqrt{\varepsilon^2 - x^2}, & -\varepsilon This is continuous and compactly supported, but not a mollifier because it is not smooth.
Semigroups Approximations to the delta functions often arise as convolution
semigroups. This amounts to the further constraint that the convolution of with must satisfy \eta_\varepsilon * \eta_\delta = \eta_{\varepsilon+\delta} for all . Convolution semigroups in that approximate the delta function are always an approximation to the identity in the above sense, however the semigroup condition is quite a strong restriction. In practice, semigroups approximating the delta function arise as
fundamental solutions or
Green's functions to physically motivated
elliptic or
parabolic partial differential equations. In the context of
applied mathematics, semigroups arise as the output of a
linear time-invariant system. Abstractly, if
A is a linear operator acting on functions of
x, then a convolution semigroup arises by solving the
initial value problem \begin{cases} \dfrac{\partial}{\partial t}\eta(t,x) = A\eta(t,x), \quad t>0 \\[5pt] \displaystyle\lim_{t\to 0^+} \eta(t,x) = \delta(x) \end{cases} in which the limit is as usual understood in the weak sense. Setting gives the associated approximate delta function. Some examples of physically important convolution semigroups arising from such a fundamental solution include the following.
The heat kernel The
heat kernel, defined by \eta_\varepsilon(x) = \frac{1}{\sqrt{2\pi\varepsilon}} \mathrm{e}^{-\frac{x^2}{2\varepsilon}} represents the temperature in an infinite wire at time , if a unit of heat energy is stored at the origin of the wire at time . This semigroup evolves according to the one-dimensional
heat equation: \frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2}. s) of the sequence of zero-centered
normal distributions \delta_a(x) = \frac{1}{\left|a\right| \sqrt{\pi}} e^{-(x/a)^2} In
probability theory, is a
normal distribution of
variance and mean . It represents the
probability density at time of the position of a particle starting at the origin following a standard
Brownian motion. In this context, the semigroup condition is then an expression of the
Markov property of Brownian motion. In higher-dimensional Euclidean space , the heat kernel is \eta_\varepsilon = \frac{1}{(2\pi\varepsilon)^{n/2}}\mathrm{e}^{-\frac{x\cdot x}{2\varepsilon}}, and has the same physical interpretation, . It also represents an approximation to the delta function in the sense that in the distribution sense as .
The Poisson kernel The
Poisson kernel \eta_\varepsilon(x) = \frac{1}{\pi}\mathrm{Im}\left\{\frac{1}{x-\mathrm{i}\varepsilon}\right\}=\frac{1}{\pi} \frac{\varepsilon}{\varepsilon^2 + x^2}=\frac{1}{2\pi}\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i} \xi x-|\varepsilon \xi|}\,d\xi is the fundamental solution of the
Laplace equation in the upper half-plane. It represents the
electrostatic potential in a semi-infinite plate whose potential along the edge is held at fixed at the delta function. The Poisson kernel is also closely related to the
Cauchy distribution and
Epanechnikov and Gaussian kernel functions. This semigroup evolves according to the equation \frac{\partial u}{\partial t} = -\left (-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}u(t,x) where the operator is rigorously defined as the
Fourier multiplier \mathcal{F}\left[\left(-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}f\right](\xi) = |2\pi\xi|\mathcal{F}f(\xi).
Oscillatory integrals In areas of physics such as
wave propagation and
wave mechanics, the equations involved are
hyperbolic and so may have more singular solutions. As a result, the approximate delta functions that arise as fundamental solutions of the associated
Cauchy problems are generally
oscillatory integrals. An example, which comes from a solution of the
Euler–Tricomi equation of
transonic gas dynamics, is the rescaled
Airy function \varepsilon^{-1/3}\operatorname{Ai}\left (x\varepsilon^{-1/3} \right). Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense. Many approximate delta functions constructed as oscillatory integrals only converge in the sense of distributions (an example is the
Dirichlet kernel below), rather than in the sense of measures. Another example is the Cauchy problem for the
wave equation in : \begin{align} c^{-2}\frac{\partial^2u}{\partial t^2} - \Delta u &= 0\\ u=0,\quad \frac{\partial u}{\partial t} = \delta &\qquad \text{for }t=0. \end{align} The solution represents the displacement from equilibrium of an infinite elastic string, with an initial disturbance at the origin. Other approximations to the identity of this kind include the
sinc function (used widely in electronics and telecommunications) \eta_\varepsilon(x)=\frac{1}{\pi x}\sin\left(\frac{x}{\varepsilon}\right)=\frac{1}{2\pi}\int_{-\frac{1}{\varepsilon}}^{\frac{1}{\varepsilon}} \cos(kx)\,dk and the
Bessel function \eta_\varepsilon(x) = \frac{1}{\varepsilon}J_{\frac{1}{\varepsilon}} \left(\frac{x+1}{\varepsilon}\right).
Plane wave decomposition One approach to the study of a linear partial differential equation L[u]=f, where is a
differential operator on , is to seek first a fundamental solution, which is a solution of the equation L[u]=\delta. When is particularly simple, this problem can often be resolved using the Fourier transform directly (as in the case of the Poisson kernel and heat kernel already mentioned). For more complicated operators, it is sometimes easier first to consider an equation of the form L[u]=h where is a
plane wave function, meaning that it has the form h = h(x\cdot\xi) for some vector . Such an equation can be resolved (if the coefficients of are
analytic functions) by the
Cauchy–Kovalevskaya theorem or (if the coefficients of are constant) by quadrature. So, if the delta function can be decomposed into plane waves, then one can in principle solve linear partial differential equations. Such a decomposition of the delta function into plane waves was part of a general technique first introduced essentially by
Johann Radon, and then developed in this form by
Fritz John (
1955). Choose so that is an even integer, and for a real number , put g(s) = \operatorname{Re}\left[\frac{-s^k\log(-is)}{k!(2\pi i)^n}\right] =\begin{cases} \frac{k!(2\pi i)^n}&n \text{ even.} \end{cases} Then is obtained by applying a power of the
Laplacian to the integral with respect to the unit
sphere measure of for in the
unit sphere : \delta(x) = \Delta_x^{(n+k)/2} \int_{S^{n-1}} g(x\cdot\xi)\,d\omega_\xi. The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function , \varphi(x) = \int_{\mathbf{R}^n}\varphi(y)\,dy\,\Delta_x^{\frac{n+k}{2}} \int_{S^{n-1}} g((x-y)\cdot\xi)\,d\omega_\xi. The result follows from the formula for the
Newtonian potential (the fundamental solution of Poisson's equation). This is essentially a form of the inversion formula for the
Radon transform because it recovers the value of from its integrals over hyperplanes. For instance, if is odd and , then the integral on the right hand side is \begin{align} & c_n \Delta^{\frac{n+1}{2}}_x\iint_{S^{n-1}} \varphi(y)|(y-x) \cdot \xi| \, d\omega_\xi \, dy \\[5pt] & \qquad = c_n \Delta^{(n+1)/2}_x \int_{S^{n-1}} \, d\omega_\xi \int_{-\infty}^\infty |p| R\varphi(\xi,p+x\cdot\xi)\,dp \end{align} where is the Radon transform of : R\varphi(\xi,p) = \int_{x\cdot\xi=p} f(x)\,d^{n-1}x. An alternative equivalent expression of the plane wave decomposition is: \delta(x) = \begin{cases} \frac{(n-1)!}{(2\pi i)^n}\displaystyle\int_{S^{n-1}}(x\cdot\xi)^{-n} \, d\omega_\xi & n\text{ even} \\ \frac{1}{2(2\pi i)^{n-1}}\displaystyle\int_{S^{n-1}}\delta^{(n-1)}(x\cdot\xi)\,d\omega_\xi & n\text{ odd}. \end{cases}
Fourier transform The delta function is a
tempered distribution, and therefore it has a well-defined
Fourier transform. Formally, one finds \widehat{\delta}(\xi)=\int_{-\infty}^\infty e^{-2\pi i x \xi} \,\delta(x)dx = 1. Properly speaking, the Fourier transform of a distribution is defined by imposing
self-adjointness of the Fourier transform under the
duality pairing \langle\cdot,\cdot\rangle of tempered distributions with
Schwartz functions. Thus \widehat{\delta} is defined as the unique tempered distribution satisfying \langle\widehat{\delta},\varphi\rangle = \langle\delta,\widehat{\varphi}\rangle for all Schwartz functions . And indeed it follows from this that \widehat{\delta}=1. As a result of this identity, the
convolution of the delta function with any other tempered distribution is simply : S*\delta = S. That is to say that is an
identity element for the convolution on tempered distributions, and in fact, the space of compactly supported distributions under convolution is an
associative algebra with identity the delta function. This property is fundamental in
signal processing, as convolution with a tempered distribution is a
linear time-invariant system, and applying the linear time-invariant system measures its
impulse response. The impulse response can be computed to any desired degree of accuracy by choosing a suitable approximation for , and once it is known, it characterizes the system completely. See . The inverse Fourier transform of the tempered distribution is the delta function. Formally, this is expressed as \int_{-\infty}^\infty 1 \cdot e^{2\pi i x\xi}\,d\xi = \delta(x) and more rigorously, it follows since \langle 1, \widehat{f}\rangle = f(0) = \langle\delta,f\rangle for all Schwartz functions . In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on . Formally, one has \int_{-\infty}^\infty e^{i 2\pi \xi_1 t} \left[e^{i 2\pi \xi_2 t}\right]^*\,dt = \int_{-\infty}^\infty e^{i 2\pi (\xi_1 - \xi_2) t} \,dt = \delta(\xi_1 - \xi_2). This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution f(t) = e^{i2\pi\xi_1 t} is \widehat{f}(\xi_2) = \delta(\xi_1-\xi_2) which again follows by imposing self-adjointness of the Fourier transform. By
analytic continuation of the Fourier transform, the
Laplace transform of the delta function is found to be \int_{0}^{\infty}\delta(t-a)\,e^{-st} \, dt=e^{-sa}.
Fourier kernels In the study of
Fourier series, a major question consists of determining whether and in what sense the Fourier series associated with a
periodic function converges to the function. The -th partial sum of the Fourier series of a function of period is defined by convolution (on the interval ) with the
Dirichlet kernel: D_N(x) = \sum_{n=-N}^N e^{inx} = \frac{\sin\left(\left(N+\frac12\right)x\right)}{\sin(x/2)}. Thus, s_N(f)(x) = D_N*f(x) = \sum_{n=-N}^N a_n e^{inx} where a_n = \frac{1}{2\pi}\int_{-\pi}^\pi f(y)e^{-iny}\,dy. A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval tends to a multiple of the delta function as . This is interpreted in the distribution sense, that s_N(f)(0) = \int_{-\pi}^{\pi} D_N(x)f(x)\,dx \to 2\pi f(0) for every compactly supported function . Thus, formally one has \delta(x) = \frac1{2\pi} \sum_{n=-\infty}^\infty e^{inx} on the interval . Despite this, the result does not hold for all compactly supported functions: that is does not converge weakly in the sense of measures. The lack of convergence of the Fourier series has led to the introduction of a variety of
summability methods to produce convergence. The method of
Cesàro summation leads to the
Fejér kernel F_N(x) = \frac1N\sum_{n=0}^{N-1} D_n(x) = \frac{1}{N}\left(\frac{\sin \frac{Nx}{2}}{\sin \frac{x}{2}}\right)^2. The
Fejér kernels tend to the delta function in a stronger sense that \int_{-\pi}^{\pi} F_N(x)f(x)\,dx \to 2\pi f(0) for every compactly supported function . The implication is that the Fourier series of any continuous function is Cesàro summable to the value of the function at every point.
Hilbert space theory The Dirac delta distribution is a
densely defined unbounded linear functional on the
Hilbert space L2 of
square-integrable functions. Indeed, smooth compactly supported functions are
dense in , and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of and to give a stronger
topology on which the delta function defines a
bounded linear functional.
Sobolev spaces The
Sobolev embedding theorem for
Sobolev spaces on the real line implies that any square-integrable function such that \|f\|_{H^1}^2 = \int_{-\infty}^\infty |\widehat{f}(\xi)|^2 (1+|\xi|^2)\,d\xi is automatically continuous, and satisfies in particular \delta[f]=|f(0)| Thus is a bounded linear functional on the Sobolev space . Equivalently is an element of the
continuous dual space of . More generally, in dimensions, one has provided .
Spaces of holomorphic functions In
complex analysis, the delta function enters via
Cauchy's integral formula, which asserts that if is a domain in the
complex plane with smooth boundary, then f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z},\quad z\in D for all
holomorphic functions in that are continuous on the closure of . As a result, the delta function is represented in this class of holomorphic functions by the Cauchy integral: \delta_z[f] = f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z}. Moreover, let be the
Hardy space consisting of the closure in of all holomorphic functions in continuous up to the boundary of . Then functions in uniquely extend to holomorphic functions in , and the Cauchy integral formula continues to hold. In particular for , the delta function is a continuous linear functional on . This is a special case of the situation in
several complex variables in which, for smooth domains , the
Szegő kernel plays the role of the Cauchy integral. Another representation of the delta function in a space of holomorphic functions is on the space H(D)\cap L^2(D) of square-integrable holomorphic functions in an open set D\subset\mathbb C^n. This is a closed subspace of L^2(D), and therefore is a Hilbert space. On the other hand, the functional that evaluates a holomorphic function in H(D)\cap L^2(D) at a point z of D is a continuous functional, and so by the Riesz representation theorem, is represented by integration against a kernel K_z(\zeta), the
Bergman kernel. This kernel is the analog of the delta function in this Hilbert space. A Hilbert space having such a kernel is called a
reproducing kernel Hilbert space. In the special case of the unit disc, one has \delta_w[f] = f(w) = \frac1\pi\iint_{|z|
Resolutions of the identity Given a complete
orthonormal basis set of functions in a separable Hilbert space, for example, the normalized
eigenvectors of a
compact self-adjoint operator, any vector can be expressed as f = \sum_{n=1}^\infty \alpha_n \varphi_n. The coefficients {αn} are found as \alpha_n = \langle \varphi_n, f \rangle, which may be represented by the notation: \alpha_n = \varphi_n^\dagger f, a form of the
bra–ket notation of Dirac. Adopting this notation, the expansion of takes the
dyadic form: f = \sum_{n=1}^\infty \varphi_n \left ( \varphi_n^\dagger f \right). Letting denote the
identity operator on the Hilbert space, the expression I = \sum_{n=1}^\infty \varphi_n \varphi_n^\dagger, is called a
resolution of the identity. When the Hilbert space is the space of square-integrable functions on a domain , the quantity: \varphi_n \varphi_n^\dagger, is an integral operator, and the expression for can be rewritten f(x) = \sum_{n=1}^\infty \int_D\, \left( \varphi_n (x) \varphi_n^*(\xi)\right) f(\xi) \, d \xi. The right-hand side converges to in the sense. It need not hold in a pointwise sense, even when is a continuous function. Nevertheless, it is common to abuse notation and write f(x) = \int \, \delta(x-\xi) f (\xi)\, d\xi, resulting in the representation of the delta function: \delta(x-\xi) = \sum_{n=1}^\infty \varphi_n (x) \varphi_n^*(\xi). With a suitable
rigged Hilbert space where contains all compactly supported smooth functions, this summation may converge in , depending on the properties of the basis . In most cases of practical interest, the orthonormal basis comes from an integral or differential operator (e.g. the
heat kernel), in which case the series converges in the
distribution sense.
Infinitesimal delta functions Cauchy used an infinitesimal to write down a unit impulse, infinitely tall and narrow Dirac-type delta function satisfying \int F(x)\delta_\alpha(x) \,dx = F(0) in a number of articles in 1827. Cauchy defined an infinitesimal in ''
Cours d'Analyse'' (1827) in terms of a sequence tending to zero. Namely, such a null sequence becomes an infinitesimal in Cauchy's and
Lazare Carnot's terminology.
Non-standard analysis allows one to rigorously treat infinitesimals. The article by contains a bibliography on modern Dirac delta functions in the context of an infinitesimal-enriched continuum provided by the
hyperreals. Here the Dirac delta can be given by an actual function, having the property that for every real function one has \int F(x)\delta_\alpha(x) \, dx = F(0) as anticipated by Fourier and Cauchy. == Dirac comb ==