Английская Википедия:Dirac delta function

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Файл:Dirac distribution PDF.svg

Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.

Файл:Dirac function approximation.gif

\sqrt{\pi}} e^{-(x/a)^2}</math>

In mathematical analysis, the Dirac delta function (or Шаблон:Mvar distribution), also known as the unit impulse,Шаблон:Sfn is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.Шаблон:SfnШаблон:SfnШаблон:Sfn Since there is no function having this property, to model the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions.

The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1. The mathematical rigor of the delta function was disputed until Laurent Schwartz developed the theory of distributions, where it is defined as a linear form acting on functions.

Motivation and overview

The graph of the Dirac delta is usually thought of as following the whole x-axis and the positive y-axis.Шаблон:Rp The Dirac delta is used to model a tall narrow spike function (an impulse), and other similar abstractions such as a point charge, point mass or electron point. For example, to calculate the dynamics of a billiard ball being struck, one can approximate the force of the impact by a Dirac delta. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the ball, by only considering the total impulse of the collision, without a detailed model of all of the elastic energy transfer at subatomic levels (for instance).

To be specific, suppose that a billiard ball is at rest. At time <math>t=0</math> it is struck by another ball, imparting it with a momentum Шаблон:Mvar, with units kg⋅m⋅s⁻¹. The exchange of momentum is not actually instantaneous, being mediated by elastic processes at the molecular and subatomic level, but for practical purposes it is convenient to consider that energy transfer as effectively instantaneous. The force therefore is Шаблон:Math; the units of Шаблон:Math are s⁻¹.

To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval Шаблон:Nowrap That is,

<math display="block">F_{\Delta t}(t) = \begin{cases} P/\Delta t& 0<t\leq T, \\ 0 &\text{otherwise}. \end{cases}</math>

Then the momentum at any time Шаблон:Mvar is found by integration:

<math display="block">p(t) = \int_0^t F_{\Delta t}(\tau)\,d\tau = \begin{cases} P & t \ge T\\ P\,t/\Delta t & 0 \le t \le T\\ 0&\text{otherwise.}\end{cases}</math>

Now, the model situation of an instantaneous transfer of momentum requires taking the limit as Шаблон:Math, giving a result everywhere except at Шаблон:Math:

<math display="block">p(t)=\begin{cases}P & t > 0\\ 0 & t < 0.\end{cases}</math>

Here the functions <math>F_{\Delta t}</math> are thought of as useful approximations to the idea of instantaneous transfer of momentum.

The delta function allows us to construct an idealized limit of these approximations. Unfortunately, the actual limit of the functions (in the sense of pointwise convergence) <math display="inline">\lim_{\Delta t\to 0^+}F_{\Delta t}</math> is zero everywhere but a single point, where it is infinite. To make proper sense of the Dirac delta, we should instead insist that the property

<math display="block">\int_{-\infty}^\infty F_{\Delta t}(t)\,dt = P,</math>

which holds for all Шаблон:Nowrap should continue to hold in the limit. So, in the equation Шаблон:Nowrap it is understood that the limit is always taken Шаблон:Em.

In applied mathematics, as we have done here, the delta function is often manipulated as a kind of limit (a weak limit) of a sequence of functions, each member of which has a tall spike at the origin: for example, a sequence of Gaussian distributions centered at the origin with variance tending to zero.

The Dirac delta is not truly a function, at least not a usual one with domain and range in real numbers. For example, the objects Шаблон:Math and Шаблон:Math are equal everywhere except at Шаблон:Math yet have integrals that are different. According to Lebesgue integration theory, if Шаблон:Mvar and Шаблон:Mvar are functions such that Шаблон:Math almost everywhere, then Шаблон:Mvar is integrable if and only if Шаблон:Mvar is integrable and the integrals of Шаблон:Mvar and Шаблон:Mvar are identical. A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right requires measure theory or the theory of distributions.

History

Joseph Fourier presented what is now called the Fourier integral theorem in his treatise Théorie analytique de la chaleur in the form:

<math display="block">f(x)=\frac{1}{2\pi}\int_{-\infty}^\infty\ \ d\alpha \, f(\alpha) \ \int_{-\infty}^\infty dp\ \cos (px-p\alpha)\ , </math>

which is tantamount to the introduction of the Шаблон:Mvar-function in the form:

<math display="block">\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty dp\ \cos (px-p\alpha) \ . </math>

Later, Augustin Cauchy expressed the theorem using exponentials:

<math display="block">f(x)=\frac{1}{2\pi} \int_{-\infty} ^ \infty \ e^{ipx}\left(\int_{-\infty}^\infty e^{-ip\alpha }f(\alpha)\,d \alpha \right) \,dp. </math>

Cauchy pointed out that in some circumstances the order of integration is significant in this result (contrast Fubini's theorem).

As justified using the theory of distributions, the Cauchy equation can be rearranged to resemble Fourier's original formulation and expose the δ-function as

<math display="block">\begin{align} f(x)&=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ipx}\left(\int_{-\infty}^\infty e^{-ip\alpha }f(\alpha)\,d \alpha \right) \,dp \\[4pt] &=\frac{1}{2\pi} \int_{-\infty}^\infty \left(\int_{-\infty}^\infty e^{ipx} e^{-ip\alpha } \,dp \right)f(\alpha)\,d \alpha =\int_{-\infty}^\infty \delta (x-\alpha) f(\alpha) \,d \alpha, \end{align}</math>

where the δ-function is expressed as

<math display="block">\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ip(x-\alpha)}\,dp \ . </math>

A rigorous interpretation of the exponential form and the various limitations upon the function f necessary for its application extended over several centuries. The problems with a classical interpretation are explained as follows:

The greatest drawback of the classical Fourier transformation is a rather narrow class of functions (originals) for which it can be effectively computed. Namely, it is necessary that these functions decrease sufficiently rapidly to zero (in the neighborhood of infinity) to ensure the existence of the Fourier integral. For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense. The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed and this removed many obstacles.

Further developments included generalization of the Fourier integral, "beginning with Plancherel's pathbreaking L²-theory (1910), continuing with Wiener's and Bochner's works (around 1930) and culminating with the amalgamation into L. Schwartz's theory of distributions (1945) ...", and leading to the formal development of the Dirac delta function.

An infinitesimal formula for an infinitely tall, unit impulse delta function (infinitesimal version of Cauchy distribution) explicitly appears in an 1827 text of Augustin Louis Cauchy. Шаблон:Sfn Siméon Denis Poisson considered the issue in connection with the study of wave propagation as did Gustav Kirchhoff somewhat later. Kirchhoff and Hermann von Helmholtz also introduced the unit impulse as a limit of Gaussians, which also corresponded to Lord Kelvin's notion of a point heat source. At the end of the 19th century, Oliver Heaviside used formal Fourier series to manipulate the unit impulse. The Dirac delta function as such was introduced by Paul Dirac in his 1927 paper The Physical Interpretation of the Quantum Dynamics and used in his textbook The Principles of Quantum Mechanics.Шаблон:Sfn He called it the "delta function" since he used it as a continuous analogue of the discrete Kronecker delta.

Definitions

The Dirac delta function <math>\delta (x)</math> can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,

<math display="block">\delta(x) \simeq \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}</math>

and which is also constrained to satisfy the identityШаблон:Sfn

<math display="block">\int_{-\infty}^\infty \delta(x) \, dx = 1.</math>

This is merely a heuristic characterization. The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties.Шаблон:Sfn

As a measure

One way to rigorously capture the notion of the Dirac delta function is to define a measure, called Dirac measure, which accepts a subset Шаблон:Mvar of the real line Шаблон:Math as an argument, and returns Шаблон:Math if Шаблон:Math, and Шаблон:Math otherwise. If the delta function is conceptualized as modeling an idealized point mass at 0, then Шаблон:Math represents the mass contained in the set Шаблон:Mvar. One may then define the integral against Шаблон:Mvar as the integral of a function against this mass distribution. Formally, the Lebesgue integral provides the necessary analytic device. The Lebesgue integral with respect to the measure Шаблон:Mvar satisfies

<math display="block">\int_{-\infty}^\infty f(x) \, \delta(dx) = f(0)</math>

for all continuous compactly supported functions Шаблон:Mvar. The measure Шаблон:Mvar is not absolutely continuous with respect to the Lebesgue measure—in fact, it is a singular measure. Consequently, the delta measure has no Radon–Nikodym derivative (with respect to Lebesgue measure)—no true function for which the property

<math display="block">\int_{-\infty}^\infty f(x)\, \delta(x)\, dx = f(0)</math>

holds.Шаблон:Sfn As a result, the latter notation is a convenient abuse of notation, and not a standard (Riemann or Lebesgue) integral.

As a probability measure on Шаблон:Math, the delta measure is characterized by its cumulative distribution function, which is the unit step function.

<math display="block">H(x) = \begin{cases} 1 & \text{if } x\ge 0\\ 0 & \text{if } x < 0. \end{cases}</math>

This means that Шаблон:Math is the integral of the cumulative indicator function Шаблон:Math with respect to the measure Шаблон:Mvar; to wit,

<math display="block">H(x) = \int_{\mathbf{R}}\mathbf{1}_{(-\infty,x]}(t)\,\delta(dt) = \delta(-\infty,x],</math>

the latter being the measure of this interval; more formally, Шаблон:Math. Thus in particular the integration of the delta function against a continuous function can be properly understood as a Riemann–Stieltjes integral:Шаблон:Sfn

<math display="block">\int_{-\infty}^\infty f(x)\,\delta(dx) = \int_{-\infty}^\infty f(x) \,dH(x).</math>

All higher moments of Шаблон:Mvar are zero. In particular, characteristic function and moment generating function are both equal to one.

As a distribution

In the theory of distributions, a generalized function is considered not a function in itself but only through how it affects other functions when "integrated" against them.Шаблон:Sfn In keeping with this philosophy, to define the delta function properly, it is enough to say what the "integral" of the delta function is against a sufficiently "good" test function Шаблон:Mvar. Test functions are also known as bump functions. If the delta function is already understood as a measure, then the Lebesgue integral of a test function against that measure supplies the necessary integral.

A typical space of test functions consists of all smooth functions on Шаблон:Math with compact support that have as many derivatives as required. As a distribution, the Dirac delta is a linear functional on the space of test functions and is defined byШаблон:Sfn

for every test function Шаблон:Mvar.

For Шаблон:Mvar to be properly a distribution, it must be continuous in a suitable topology on the space of test functions. In general, for a linear functional Шаблон:Mvar on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer Шаблон:Mvar there is an integer Шаблон:Math and a constant Шаблон:Mvar such that for every test function Шаблон:Mvar, one has the inequalityШаблон:Sfn

<math display="block">\left|S[\varphi]\right| \le C_N \sum_{k=0}^{M_N}\sup_{x\in [-N,N]} \left|\varphi^{(k)}(x)\right|</math>

where Шаблон:Math represents the supremum. With the Шаблон:Mvar distribution, one has such an inequality (with Шаблон:Math with Шаблон:Math for all Шаблон:Mvar. Thus Шаблон:Mvar is a distribution of order zero. It is, furthermore, a distribution with compact support (the support being Шаблон:Math).

The delta distribution can also be defined in several equivalent ways. For instance, it is the distributional derivative of the Heaviside step function. This means that for every test function Шаблон:Mvar, one has

<math display="block">\delta[\varphi] = -\int_{-\infty}^\infty \varphi'(x)\,H(x)\,dx.</math>

Intuitively, if integration by parts were permitted, then the latter integral should simplify to

<math display="block">\int_{-\infty}^\infty \varphi(x)\,H'(x)\,dx = \int_{-\infty}^\infty \varphi(x)\,\delta(x)\,dx,</math>

and indeed, a form of integration by parts is permitted for the Stieltjes integral, and in that case, one does have

<math display="block">-\int_{-\infty}^\infty \varphi'(x)\,H(x)\, dx = \int_{-\infty}^\infty \varphi(x)\,dH(x).</math>

In the context of measure theory, the Dirac measure gives rise to distribution by integration. Conversely, equation (Шаблон:EquationNote) defines a Daniell integral on the space of all compactly supported continuous functions Шаблон:Mvar which, by the Riesz representation theorem, can be represented as the Lebesgue integral of Шаблон:Mvar with respect to some Radon measure.

Generally, when the term Dirac delta function is used, it is in the sense of distributions rather than measures, the Dirac measure being among several terms for the corresponding notion in measure theory. Some sources may also use the term Dirac delta distribution.

Generalizations

The delta function can be defined in Шаблон:Mvar-dimensional Euclidean space Шаблон:Math as the measure such that

<math display="block">\int_{\mathbf{R}^n} f(\mathbf{x})\,\delta(d\mathbf{x}) = f(\mathbf{0})</math>

for every compactly supported continuous function Шаблон:Mvar. As a measure, the Шаблон:Mvar-dimensional delta function is the product measure of the 1-dimensional delta functions in each variable separately. Thus, formally, with Шаблон:Math, one hasШаблон:Sfn

The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case.Шаблон:Sfn However, despite widespread use in engineering contexts, (Шаблон:EquationNote) should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.Шаблон:SfnШаблон:Sfn

The notion of a Dirac measure makes sense on any set.Шаблон:Sfn Thus if Шаблон:Mvar is a set, Шаблон:Math is a marked point, and Шаблон:Math is any sigma algebra of subsets of Шаблон:Mvar, then the measure defined on sets Шаблон:Math by

<math display="block">\delta_{x_0}(A)=\begin{cases} 1 &\text{if }x_0\in A\\ 0 &\text{if }x_0\notin A \end{cases}</math>

is the delta measure or unit mass concentrated at Шаблон:Math.

Another common generalization of the delta function is to a differentiable manifold where most of its properties as a distribution can also be exploited because of the differentiable structure. The delta function on a manifold Шаблон:Mvar centered at the point Шаблон:Math is defined as the following distribution:

for all compactly supported smooth real-valued functions Шаблон:Mvar on Шаблон:Mvar.Шаблон:Sfn A common special case of this construction is a case in which Шаблон:Mvar is an open set in the Euclidean space Шаблон:Math.

On a locally compact Hausdorff space Шаблон:Mvar, the Dirac delta measure concentrated at a point Шаблон:Mvar is the Radon measure associated with the Daniell integral (Шаблон:EquationNote) on compactly supported continuous functions Шаблон:Mvar. At this level of generality, calculus as such is no longer possible, however a variety of techniques from abstract analysis are available. For instance, the mapping <math>x_0\mapsto \delta_{x_0}</math> is a continuous embedding of Шаблон:Mvar into the space of finite Radon measures on Шаблон:Mvar, equipped with its vague topology. Moreover, the convex hull of the image of Шаблон:Mvar under this embedding is dense in the space of probability measures on Шаблон:Mvar.Шаблон:Sfn

Properties

Scaling and symmetry

The delta function satisfies the following scaling property for a non-zero scalar Шаблон:Mvar:Шаблон:Sfn

<math display="block">\int_{-\infty}^\infty \delta(\alpha x)\,dx =\int_{-\infty}^\infty \delta(u)\,\frac{du}{|\alpha|} =\frac{1}{|\alpha|}</math>

and so Шаблон:NumBlk

Scaling property proof: <math display="block">\int\limits_{-\infty}^{\infty} dx\ g(x) \delta (ax) = \frac{1}{a}\int\limits_{-\infty}^{\infty} dx'\ g\left(\frac{x'}{a}\right) \delta (x') </math> where a change of variable Шаблон:Math is used. If Шаблон:Mvar is negative, i.e., Шаблон:Math, then <math display="block">\int\limits_{-\infty}^{\infty} dx\ g(x) \delta (ax) = \frac{1}{-\left \vert a \right \vert}\int\limits_{\infty}^{-\infty} dx'\ g\left(\frac{x'}{a}\right) \delta (x') = \frac{1}{\left \vert a \right \vert}\int\limits_{-\infty}^{\infty} dx'\ g\left(\frac{x'}{a}\right) \delta (x') = \frac{1}{\left \vert a \right \vert}g(0). </math> Thus, Шаблон:Nowrap

In particular, the delta function is an even distribution (symmetry), in the sense that

<math display="block">\delta(-x) = \delta(x)</math>

which is homogeneous of degree Шаблон:Math.

Algebraic properties

The distributional product of Шаблон:Mvar with Шаблон:Mvar is equal to zero:

<math display="block">x\,\delta(x) = 0.</math>

More generally, <math>(x-a)^n\delta(x-a) =0</math> for all positive integers <math>n</math>.

Conversely, if Шаблон:Math, where Шаблон:Mvar and Шаблон:Mvar are distributions, then

<math display="block">f(x) = g(x) +c \delta(x)</math>

for some constant Шаблон:Mvar.Шаблон:Sfn

Translation

The integral of the time-delayed Dirac delta is

<math display="block">\int_{-\infty}^\infty f(t) \,\delta(t-T)\,dt = f(T).</math>

This is sometimes referred to as the sifting property or the sampling property. The delta function is said to "sift out" the value of f(t) at t = T.

It follows that the effect of convolving a function Шаблон:Math with the time-delayed Dirac delta <math> \delta_T(t) = \delta(t-T)</math> is to time-delay Шаблон:Math by the same amount:

<math display="block">\begin{align} (f * \delta_T)(t) \ &\stackrel{\mathrm{def}}{=}\ \int_{-\infty}^\infty f(\tau)\, \delta(t-T-\tau) \, d\tau \\ &= \int_{-\infty}^\infty f(\tau) \,\delta(\tau-(t-T)) \,d\tau \qquad \text{since}~ \delta(-x) = \delta(x) ~~ \text{by (4)}\\ &= f(t-T). \end{align}</math>

The sifting property holds under the precise condition that Шаблон:Mvar be a tempered distribution (see the discussion of the Fourier transform below). As a special case, for instance, we have the identity (understood in the distribution sense)

<math display="block">\int_{-\infty}^\infty \delta (\xi-x) \delta(x-\eta) \,dx = \delta(\eta-\xi).</math>

Composition with a function

More generally, the delta distribution may be composed with a smooth function Шаблон:Math in such a way that the familiar change of variables formula holds, that

<math display="block">\int_{\R} \delta\bigl(g(x)\bigr) f\bigl(g(x)\bigr) \left|g'(x)\right| dx = \int_{g(\R)} \delta(u)\,f(u)\,du</math>

provided that Шаблон:Mvar is a continuously differentiable function with Шаблон:Math nowhere zero.Шаблон:Sfn That is, there is a unique way to assign meaning to the distribution <math>\delta\circ g</math> so that this identity holds for all compactly supported test functions Шаблон:Mvar. Therefore, the domain must be broken up to exclude the Шаблон:Math point. This distribution satisfies Шаблон:Math if Шаблон:Mvar is nowhere zero, and otherwise if Шаблон:Mvar has a real root at Шаблон:Math, then

<math display="block">\delta(g(x)) = \frac{\delta(x-x_0)}{|g'(x_0)|}.</math>

It is natural therefore to Шаблон:Em the composition Шаблон:Math for continuously differentiable functions Шаблон:Mvar by

<math display="block">\delta(g(x)) = \sum_i \frac{\delta(x-x_i)}{|g'(x_i)|}</math>

where the sum extends over all roots (i.e., all the different ones) of Шаблон:Mvar, which are assumed to be simple. Thus, for example

<math display="block">\delta\left(x^2-\alpha^2\right) = \frac{1}{2|\alpha|} \Big[\delta\left(x+\alpha\right)+\delta\left(x-\alpha\right)\Big].</math>

In the integral form, the generalized scaling property may be written as

<math display="block"> \int_{-\infty}^\infty f(x) \, \delta(g(x)) \, dx = \sum_{i}\frac{f(x_i)}{|g'(x_i)|}. </math>

Indefinite integral

For a constant <math>a \isin \mathbb{R}</math> and a "well-behaved" arbitrary real-valued function Шаблон:Math, <math display="block">\displaystyle{\int}y(x)\delta(x-a)dx = y(a)H(x-a) + c,</math> where Шаблон:Math is the Heaviside step function and Шаблон:Math is an integration constant.

Properties in n dimensions

The delta distribution in an Шаблон:Mvar-dimensional space satisfies the following scaling property instead, <math display="block">\delta(\alpha\mathbf{x}) = |\alpha|^{-n}\delta(\mathbf{x}) ~,</math> so that Шаблон:Mvar is a homogeneous distribution of degree Шаблон:Math.

Under any reflection or rotation Шаблон:Mvar, the delta function is invariant, <math display="block">\delta(\rho \mathbf{x}) = \delta(\mathbf{x})~.</math>

As in the one-variable case, it is possible to define the composition of Шаблон:Mvar with a bi-Lipschitz function Шаблон:Math uniquely so that the identity <math display="block">\int_{\R^n} \delta(g(\mathbf{x}))\, f(g(\mathbf{x}))\left|\det g'(\mathbf{x})\right| d\mathbf{x} = \int_{g(\R^n)} \delta(\mathbf{u}) f(\mathbf{u})\,d\mathbf{u}</math> for all compactly supported functions Шаблон:Mvar.

Using the coarea formula from geometric measure theory, one can also define the composition of the delta function with a submersion from one Euclidean space to another one of different dimension; the result is a type of current. In the special case of a continuously differentiable function Шаблон:Math such that the gradient of Шаблон:Mvar is nowhere zero, the following identity holdsШаблон:Sfn <math display="block">\int_{\R^n} f(\mathbf{x}) \, \delta(g(\mathbf{x})) \,d\mathbf{x} = \int_{g^{-1}(0)}\frac{f(\mathbf{x})}{|\mathbf{\nabla}g|}\,d\sigma(\mathbf{x}) </math> where the integral on the right is over Шаблон:Math, the Шаблон:Math-dimensional surface defined by Шаблон:Math with respect to the Minkowski content measure. This is known as a simple layer integral.

More generally, if Шаблон:Mvar is a smooth hypersurface of Шаблон:Math, then we can associate to Шаблон:Mvar the distribution that integrates any compactly supported smooth function Шаблон:Mvar over Шаблон:Mvar: <math display="block">\delta_S[g] = \int_S g(\mathbf{s})\,d\sigma(\mathbf{s})</math>

where Шаблон:Mvar is the hypersurface measure associated to Шаблон:Mvar. This generalization is associated with the potential theory of simple layer potentials on Шаблон:Mvar. If Шаблон:Mvar is a domain in Шаблон:Math with smooth boundary Шаблон:Mvar, then Шаблон:Math is equal to the normal derivative of the indicator function of Шаблон:Mvar in the distribution sense,

<math display="block">-\int_{\R^n}g(\mathbf{x})\,\frac{\partial 1_D(\mathbf{x})}{\partial n}\,d\mathbf{x}=\int_S\,g(\mathbf{s})\, d\sigma(\mathbf{s}),</math>

where Шаблон:Mvar is the outward normal.Шаблон:SfnШаблон:Sfn For a proof, see e.g. the article on the surface delta function.

In three dimensions, the delta function is represented in spherical coordinates by:

<math display="block">\delta(\mathbf{r}-\mathbf{r}_0) = \begin{cases}

   \displaystyle\frac{1}{r^2\sin\theta}\delta(r-r_0) \delta(\theta-\theta_0)\delta(\phi-\phi_0)& x_0,y_0,z_0 \ne 0 \\
   \displaystyle\frac{1}{2\pi r^2\sin\theta}\delta(r-r_0) \delta(\theta-\theta_0)& x_0=y_0=0,\ z_0 \ne 0 \\ 
   \displaystyle\frac{1}{4\pi r^2}\delta(r-r_0) & x_0=y_0=z_0 = 0  

\end{cases}</math>

Fourier transform

The delta function is a tempered distribution, and therefore it has a well-defined Fourier transform. Formally, one finds

<math display="block">\widehat{\delta}(\xi)=\int_{-\infty}^\infty e^{-2\pi i x \xi} \,\delta(x)dx = 1.</math>

Properly speaking, the Fourier transform of a distribution is defined by imposing self-adjointness of the Fourier transform under the duality pairing <math>\langle\cdot,\cdot\rangle</math> of tempered distributions with Schwartz functions. Thus <math>\widehat{\delta}</math> is defined as the unique tempered distribution satisfying

<math display="block">\langle\widehat{\delta},\varphi\rangle = \langle\delta,\widehat{\varphi}\rangle</math>

for all Schwartz functions Шаблон:Mvar. And indeed it follows from this that <math>\widehat{\delta}=1.</math>

As a result of this identity, the convolution of the delta function with any other tempered distribution Шаблон:Mvar is simply Шаблон:Mvar:

<math display="block">S*\delta = S.</math>

That is to say that Шаблон:Mvar 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 Шаблон:Mvar, and once it is known, it characterizes the system completely. See Шаблон:Section link.

The inverse Fourier transform of the tempered distribution Шаблон:Math is the delta function. Formally, this is expressed as <math display="block">\int_{-\infty}^\infty 1 \cdot e^{2\pi i x\xi}\,d\xi = \delta(x)</math> and more rigorously, it follows since <math display="block">\langle 1, \widehat{f}\rangle = f(0) = \langle\delta,f\rangle</math> for all Schwartz functions Шаблон:Mvar.

In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on Шаблон:Math. Formally, one has <math display="block">\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_2 - \xi_1) t} \,dt = \delta(\xi_2 - \xi_1).</math>

This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution <math display="block">f(t) = e^{i2\pi\xi_1 t}</math> is <math display="block">\widehat{f}(\xi_2) = \delta(\xi_1-\xi_2)</math> 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 beШаблон:Sfn <math display="block"> \int_{0}^{\infty}\delta(t-a)\,e^{-st} \, dt=e^{-sa}.</math>

Derivatives

The derivative of the Dirac delta distribution, denoted Шаблон:Math and also called the Dirac delta prime or Dirac delta derivative as described in Laplacian of the indicator, is defined on compactly supported smooth test functions Шаблон:Mvar byШаблон:Sfn <math display="block">\delta'[\varphi] = -\delta[\varphi']=-\varphi'(0).</math>

The first equality here is a kind of integration by parts, for if Шаблон:Mvar were a true function then <math display="block">\int_{-\infty}^\infty \delta'(x)\varphi(x)\,dx = \delta(x)\varphi(x)|_{-\infty}^{\infty} -\int_{-\infty}^\infty \delta(x) \varphi'(x)\,dx = -\int_{-\infty}^\infty \delta(x) \varphi'(x)\,dx = -\varphi'(0).</math>

By mathematical induction, the Шаблон:Mvar-th derivative of Шаблон:Mvar is defined similarly as the distribution given on test functions by

<math display="block">\delta^{(k)}[\varphi] = (-1)^k \varphi^{(k)}(0).</math>

In particular, Шаблон:Mvar is an infinitely differentiable distribution.

The first derivative of the delta function is the distributional limit of the difference quotients:Шаблон:Sfn <math display="block">\delta'(x) = \lim_{h\to 0} \frac{\delta(x+h)-\delta(x)}{h}.</math>

More properly, one has <math display="block">\delta' = \lim_{h\to 0} \frac{1}{h}(\tau_h\delta - \delta)</math> where Шаблон:Mvar is the translation operator, defined on functions by Шаблон:Math, and on a distribution Шаблон:Mvar by <math display="block">(\tau_h S)[\varphi] = S[\tau_{-h}\varphi].</math>

In the theory of electromagnetism, the first derivative of the delta function represents a point magnetic dipole situated at the origin. Accordingly, it is referred to as a dipole or the doublet function.

The derivative of the delta function satisfies a number of basic properties, including:Шаблон:Sfn <math display="block"> \begin{align}

\delta'(-x) &= -\delta'(x) \\
x\delta'(x) &= -\delta(x)

\end{align} </math> which can be shown by applying a test function and integrating by parts.

The latter of these properties can also be demonstrated by applying distributional derivative definition, Liebniz 's theorem and linearity of inner product:

<math display="block"> \begin{align} \langle x\delta', \varphi \rangle \, &= \, \langle \delta', x\varphi \rangle \, = \, -\langle\delta,(x\varphi)'\rangle \, = \, - \langle \delta, x'\varphi + x\varphi'\rangle \, = \, - \langle \delta, x'\varphi\rangle - \langle\delta, x\varphi'\rangle \, = \, - x'(0)\varphi(0) - x(0)\varphi'(0) \\ &= \, -x'(0) \langle \delta , \varphi \rangle - x(0) \langle \delta, \varphi' \rangle \, = \, -x'(0) \langle \delta,\varphi\rangle + x(0) \langle \delta', \varphi \rangle \, = \, \langle x(0)\delta' - x'(0)\delta, \varphi \rangle \\

\Longrightarrow x(t)\delta'(t) &= x(0)\delta'(t) - x'(0)\delta(t) = -x'(0)\delta(t) = -\delta(t)

\end{align} </math>

Furthermore, the convolution of Шаблон:Mvar with a compactly-supported, smooth function Шаблон:Mvar is

<math display="block">\delta'*f = \delta*f' = f',</math>

which follows from the properties of the distributional derivative of a convolution.

Higher dimensions

More generally, on an open set Шаблон:Mvar in the Шаблон:Mvar-dimensional Euclidean space <math>\mathbb{R}^n</math>, the Dirac delta distribution centered at a point Шаблон:Math is defined byШаблон:Sfn <math display="block">\delta_a[\varphi]=\varphi(a)</math> for all <math>\varphi \in C_c^\infty(U)</math>, the space of all smooth functions with compact support on Шаблон:Mvar. If <math>\alpha = (\alpha_1, \ldots, \alpha_n)</math> is any multi-index with <math> |\alpha|=\alpha_1+\cdots+\alpha_n</math> and <math>\partial^\alpha</math> denotes the associated mixed partial derivative operator, then the Шаблон:Mvar-th derivative Шаблон:Mvar of Шаблон:Mvar is given byШаблон:Sfn

<math display="block">\left\langle \partial^\alpha \delta_{a}, \, \varphi \right\rangle = (-1)^{| \alpha |} \left\langle \delta_{a}, \partial^{\alpha} \varphi \right\rangle = (-1)^{| \alpha |} \partial^\alpha \varphi (x) \Big|_{x = a} \quad \text{ for all } \varphi \in C_c^\infty(U).</math>

That is, the Шаблон:Mvar-th derivative of Шаблон:Mvar is the distribution whose value on any test function Шаблон:Mvar is the Шаблон:Mvar-th derivative of Шаблон:Mvar at Шаблон:Mvar (with the appropriate positive or negative sign).

The first partial derivatives of the delta function are thought of as double layers along the coordinate planes. More generally, the normal derivative of a simple layer supported on a surface is a double layer supported on that surface and represents a laminar magnetic monopole. Higher derivatives of the delta function are known in physics as multipoles.

Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support. If Шаблон:Mvar is any distribution on Шаблон:Mvar supported on the set Шаблон:Math consisting of a single point, then there is an integer Шаблон:Mvar and coefficients Шаблон:Mvar such thatШаблон:SfnШаблон:Sfn <math display="block">S = \sum_{|\alpha|\le m} c_\alpha \partial^\alpha\delta_a.</math>

Representations of the delta function

The delta function can be viewed as the limit of a sequence of functions

<math display="block">\delta (x) = \lim_{\varepsilon\to 0^+} \eta_\varepsilon(x), </math>

where Шаблон:Math is sometimes called a nascent delta functionШаблон:Anchor. This limit is meant in a weak sense: either that

for all continuous functions Шаблон:Mvar having compact support, or that this limit holds for all smooth functions Шаблон:Mvar with compact support. The difference between these two slightly different modes of weak convergence is often subtle: the former is convergence in the vague topology of measures, and the latter is convergence in the sense of distributions.

Approximations to the identity

Typically a nascent delta function Шаблон:Mvar can be constructed in the following manner. Let Шаблон:Mvar be an absolutely integrable function on Шаблон:Math of total integral Шаблон:Math, and define <math display="block">\eta_\varepsilon(x) = \varepsilon^{-1} \eta \left (\frac{x}{\varepsilon} \right). </math>

In Шаблон:Mvar dimensions, one uses instead the scaling <math display="block">\eta_\varepsilon(x) = \varepsilon^{-n} \eta \left (\frac{x}{\varepsilon} \right). </math>

Then a simple change of variables shows that Шаблон:Mvar also has integral Шаблон:Math. One may show that (Шаблон:EquationNote) holds for all continuous compactly supported functions Шаблон:Mvar,Шаблон:Sfn and so Шаблон:Mvar converges weakly to Шаблон:Mvar in the sense of measures.

The Шаблон:Mvar constructed in this way are known as an approximation to the identity.Шаблон:Sfn This terminology is because the space Шаблон:Math of absolutely integrable functions is closed under the operation of convolution of functions: Шаблон:Math whenever Шаблон:Mvar and Шаблон:Mvar are in Шаблон:Math. However, there is no identity in Шаблон:Math for the convolution product: no element Шаблон:Mvar such that Шаблон:Math for all Шаблон:Mvar. Nevertheless, the sequence Шаблон:Mvar does approximate such an identity in the sense that

<math display="block">f*\eta_\varepsilon \to f \quad \text{as }\varepsilon\to 0.</math>

This limit holds in the sense of mean convergence (convergence in Шаблон:Math). Further conditions on the Шаблон:Mvar, for instance that it be a mollifier associated to a compactly supported function, are needed to ensure pointwise convergence almost everywhere.

If the initial Шаблон:Math is itself smooth and compactly supported then the sequence is called a mollifier. The standard mollifier is obtained by choosing Шаблон:Mvar to be a suitably normalized bump function, for instance

<math display="block">\eta(x) = \begin{cases} e^{-\frac{1}{1-|x|^2}}& \text{if } |x| < 1\\ 0 & \text{if } |x|\geq 1. \end{cases}</math>

In some situations such as numerical analysis, a piecewise linear approximation to the identity is desirable. This can be obtained by taking Шаблон:Math to be a hat function. With this choice of Шаблон:Math, one has

<math display="block"> \eta_\varepsilon(x) = \varepsilon^{-1}\max \left (1-\left|\frac{x}{\varepsilon}\right|,0 \right) </math>

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 Шаблон:Math 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 Шаблон:Math to be any probability distribution at all, and letting Шаблон:Math as above will give rise to an approximation to the identity. In general this converges more rapidly to a delta function if, in addition, Шаблон:Mvar has mean Шаблон:Math and has small higher moments. For instance, if Шаблон:Math is the uniform distribution on Шаблон:Nowrap also known as the rectangular function, then:Шаблон:Sfn <math display="block"> \eta_\varepsilon(x) = \frac{1}{\varepsilon}\operatorname{rect}\left(\frac{x}{\varepsilon}\right)= \begin{cases} \frac{1}{\varepsilon},&-\frac{\varepsilon}{2}<x<\frac{\varepsilon}{2}, \\ 0, &\text{otherwise}. \end{cases}</math>

Another example is with the Wigner semicircle distribution <math display="block">\eta_\varepsilon(x)= \begin{cases} \frac{2}{\pi \varepsilon^2}\sqrt{\varepsilon^2 - x^2}, & -\varepsilon < x < \varepsilon, \\ 0, & \text{otherwise}. \end{cases}</math>

This is continuous and compactly supported, but not a mollifier because it is not smooth.

Semigroups

Nascent delta functions often arise as convolution semigroups. This amounts to the further constraint that the convolution of Шаблон:Mvar with Шаблон:Mvar must satisfy <math display="block">\eta_\varepsilon * \eta_\delta = \eta_{\varepsilon+\delta}</math>

for all Шаблон:Math. Convolution semigroups in Шаблон:Math that form a nascent 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

<math display="block">\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}</math>

in which the limit is as usual understood in the weak sense. Setting Шаблон:Math gives the associated nascent 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

<math display="block">\eta_\varepsilon(x) = \frac{1}{\sqrt{2\pi\varepsilon}} \mathrm{e}^{-\frac{x^2}{2\varepsilon}}</math>

represents the temperature in an infinite wire at time Шаблон:Math, if a unit of heat energy is stored at the origin of the wire at time Шаблон:Math. This semigroup evolves according to the one-dimensional heat equation:

<math display="block">\frac{\partial u}{\partial t} = \frac{1}{2}\frac{\partial^2 u}{\partial x^2}.</math>

In probability theory, Шаблон:Math is a normal distribution of variance Шаблон:Mvar and mean Шаблон:Math. It represents the probability density at time Шаблон:Math 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 Шаблон:Math, the heat kernel is <math display="block">\eta_\varepsilon = \frac{1}{(2\pi\varepsilon)^{n/2}}\mathrm{e}^{-\frac{x\cdot x}{2\varepsilon}},</math> and has the same physical interpretation, Шаблон:Lang. It also represents a nascent delta function in the sense that Шаблон:Math in the distribution sense as Шаблон:Math.

The Poisson kernel

The Poisson kernel <math display="block">\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</math>

is the fundamental solution of the Laplace equation in the upper half-plane.Шаблон:Sfn 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 <math display="block">\frac{\partial u}{\partial t} = -\left (-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}u(t,x)</math>

where the operator is rigorously defined as the Fourier multiplier <math display="block">\mathcal{F}\left[\left(-\frac{\partial^2}{\partial x^2} \right)^{\frac{1}{2}}f\right](\xi) = |2\pi\xi|\mathcal{F}f(\xi).</math>

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 nascent 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,Шаблон:Sfn is the rescaled Airy function <math display="block">\varepsilon^{-1/3}\operatorname{Ai}\left (x\varepsilon^{-1/3} \right). </math>

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 nascent 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 Шаблон:Math:Шаблон:Sfn <math display="block"> \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} </math>

The solution Шаблон:Mvar 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) <math display="block">\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 </math>

and the Bessel function <math display="block"> \eta_\varepsilon(x) = \frac{1}{\varepsilon}J_{\frac{1}{\varepsilon}} \left(\frac{x+1}{\varepsilon}\right). </math>

Plane wave decomposition

One approach to the study of a linear partial differential equation <math display="block">L[u]=f,</math>

where Шаблон:Mvar is a differential operator on Шаблон:Math, is to seek first a fundamental solution, which is a solution of the equation <math display="block">L[u]=\delta.</math>

When Шаблон:Mvar 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 <math display="block">L[u]=h</math>

where Шаблон:Mvar is a plane wave function, meaning that it has the form <math display="block">h = h(x\cdot\xi)</math>

for some vector Шаблон:Mvar. Such an equation can be resolved (if the coefficients of Шаблон:Mvar are analytic functions) by the Cauchy–Kovalevskaya theorem or (if the coefficients of Шаблон:Mvar 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).Шаблон:Sfn Choose Шаблон:Mvar so that Шаблон:Math is an even integer, and for a real number Шаблон:Mvar, put <math display="block">g(s) = \operatorname{Re}\left[\frac{-s^k\log(-is)}{k!(2\pi i)^n}\right] =\begin{cases} \frac{|s|^k}{4k!(2\pi i)^{n-1}} &n \text{ odd}\\[5pt] -\frac{|s|^k\log|s|}{k!(2\pi i)^n}&n \text{ even.} \end{cases}</math>

Then Шаблон:Mvar is obtained by applying a power of the Laplacian to the integral with respect to the unit sphere measure Шаблон:Mvar of Шаблон:Math for Шаблон:Mvar in the unit sphere Шаблон:Math: <math display="block">\delta(x) = \Delta_x^{(n+k)/2} \int_{S^{n-1}} g(x\cdot\xi)\,d\omega_\xi.</math>

The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function Шаблон:Mvar, <math display="block">\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.</math>

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 Шаблон:Math from its integrals over hyperplanes. For instance, if Шаблон:Mvar is odd and Шаблон:Math, then the integral on the right hand side is <math display="block"> \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} </math>

where Шаблон:Math is the Radon transform of Шаблон:Mvar: <math display="block">R\varphi(\xi,p) = \int_{x\cdot\xi=p} f(x)\,d^{n-1}x.</math>

An alternative equivalent expression of the plane wave decomposition is:Шаблон:Sfn <math display="block">\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}</math>

Fourier kernels

Шаблон:See also 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 Шаблон:Mvar-th partial sum of the Fourier series of a function Шаблон:Mvar of period Шаблон:Math is defined by convolution (on the interval Шаблон:Closed-closed) with the Dirichlet kernel: <math display="block">D_N(x) = \sum_{n=-N}^N e^{inx} = \frac{\sin\left(\left(N+\frac12\right)x\right)}{\sin(x/2)}.</math> Thus, <math display="block">s_N(f)(x) = D_N*f(x) = \sum_{n=-N}^N a_n e^{inx}</math> where <math display="block">a_n = \frac{1}{2\pi}\int_{-\pi}^\pi f(y)e^{-iny}\,dy.</math> A fundamental result of elementary Fourier series states that the Dirichlet kernel restricted to the interval Шаблон:Closed-closed tends to a multiple of the delta function as Шаблон:Math. This is interpreted in the distribution sense, that <math display="block">s_N(f)(0) = \int_{-\pi}^{\pi} D_N(x)f(x)\,dx \to 2\pi f(0)</math> for every compactly supported Шаблон:Em function Шаблон:Mvar. Thus, formally one has <math display="block">\delta(x) = \frac1{2\pi} \sum_{n=-\infty}^\infty e^{inx}</math> on the interval Шаблон:Closed-closed.

Despite this, the result does not hold for all compactly supported Шаблон:Em functions: that is Шаблон:Math 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 kernelШаблон:Sfn

<math display="block">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.</math>

The Fejér kernels tend to the delta function in a stronger sense that

<math display="block">\int_{-\pi}^{\pi} F_N(x)f(x)\,dx \to 2\pi f(0)</math>

for every compactly supported Шаблон:Em function Шаблон:Mvar. 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 Шаблон:Math, and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of Шаблон:Math 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 Шаблон:Math implies that any square-integrable function Шаблон:Mvar such that

<math display="block">\|f\|_{H^1}^2 = \int_{-\infty}^\infty |\widehat{f}(\xi)|^2 (1+|\xi|^2)\,d\xi < \infty</math>

is automatically continuous, and satisfies in particular

<math display="block">\delta[f]=|f(0)| < C \|f\|_{H^1}.</math>

Thus Шаблон:Mvar is a bounded linear functional on the Sobolev space Шаблон:Math. Equivalently Шаблон:Mvar is an element of the continuous dual space Шаблон:Math of Шаблон:Math. More generally, in Шаблон:Mvar dimensions, one has Шаблон:Math provided Шаблон:Math.

Spaces of holomorphic functions

In complex analysis, the delta function enters via Cauchy's integral formula, which asserts that if Шаблон:Mvar is a domain in the complex plane with smooth boundary, then

<math display="block">f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z},\quad z\in D</math>

for all holomorphic functions Шаблон:Mvar in Шаблон:Mvar that are continuous on the closure of Шаблон:Mvar. As a result, the delta function Шаблон:Math is represented in this class of holomorphic functions by the Cauchy integral:

<math display="block">\delta_z[f] = f(z) = \frac{1}{2\pi i} \oint_{\partial D} \frac{f(\zeta)\,d\zeta}{\zeta-z}.</math>

Moreover, let Шаблон:Math be the Hardy space consisting of the closure in Шаблон:Math of all holomorphic functions in Шаблон:Mvar continuous up to the boundary of Шаблон:Mvar. Then functions in Шаблон:Math uniquely extend to holomorphic functions in Шаблон:Mvar, and the Cauchy integral formula continues to hold. In particular for Шаблон:Math, the delta function Шаблон:Mvar is a continuous linear functional on Шаблон:Math. This is a special case of the situation in several complex variables in which, for smooth domains Шаблон:Mvar, the Szegő kernel plays the role of the Cauchy integral.Шаблон:Sfn

Resolutions of the identity

Given a complete orthonormal basis set of functions Шаблон:Math in a separable Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector Шаблон:Mvar can be expressed as <math display="block">f = \sum_{n=1}^\infty \alpha_n \varphi_n. </math> The coefficients {α_n} are found as <math display="block">\alpha_n = \langle \varphi_n, f \rangle,</math> which may be represented by the notation: <math display="block">\alpha_n = \varphi_n^\dagger f, </math> a form of the bra–ket notation of Dirac. Adopting this notation, the expansion of Шаблон:Mvar takes the dyadic form:Шаблон:Sfn

<math display="block">f = \sum_{n=1}^\infty \varphi_n \left ( \varphi_n^\dagger f \right). </math>

Letting Шаблон:Mvar denote the identity operator on the Hilbert space, the expression

<math display="block">I = \sum_{n=1}^\infty \varphi_n \varphi_n^\dagger, </math>

is called a resolution of the identity. When the Hilbert space is the space Шаблон:Math of square-integrable functions on a domain Шаблон:Mvar, the quantity:

<math display="block">\varphi_n \varphi_n^\dagger, </math>

is an integral operator, and the expression for Шаблон:Mvar can be rewritten

<math display="block">f(x) = \sum_{n=1}^\infty \int_D\, \left( \varphi_n (x) \varphi_n^*(\xi)\right) f(\xi) \, d \xi.</math>

The right-hand side converges to Шаблон:Mvar in the Шаблон:Math sense. It need not hold in a pointwise sense, even when Шаблон:Mvar is a continuous function. Nevertheless, it is common to abuse notation and write

<math display="block">f(x) = \int \, \delta(x-\xi) f (\xi)\, d\xi, </math>

resulting in the representation of the delta function:Шаблон:Sfn

<math display="block">\delta(x-\xi) = \sum_{n=1}^\infty \varphi_n (x) \varphi_n^*(\xi). </math>

With a suitable rigged Hilbert space Шаблон:Math where Шаблон:Math contains all compactly supported smooth functions, this summation may converge in Шаблон:Math, depending on the properties of the basis Шаблон:Math. In most cases of practical interest, the orthonormal basis comes from an integral or differential operator, in which case the series converges in the distribution sense.Шаблон:Sfn

Infinitesimal delta functions

Cauchy used an infinitesimal Шаблон:Mvar to write down a unit impulse, infinitely tall and narrow Dirac-type delta function Шаблон:Mvar satisfying <math display="inline">\int F(x)\delta_\alpha(x) \,dx = F(0)</math> in a number of articles in 1827.Шаблон:Sfn 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 Шаблон:Harvtxt 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 Шаблон:Mvar one has <math display="inline">\int F(x)\delta_\alpha(x) \, dx = F(0)</math> as anticipated by Fourier and Cauchy.

Dirac comb

Файл:Dirac comb.svg

A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of Шаблон:Mvar

A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the Sha distribution, creates a sampling function, often used in digital signal processing (DSP) and discrete time signal analysis. The Dirac comb is given as the infinite sum, whose limit is understood in the distribution sense,

<math display="block">\operatorname{\text{Ш}}(x) = \sum_{n=-\infty}^\infty \delta(x-n),</math>

which is a sequence of point masses at each of the integers.

Up to an overall normalizing constant, the Dirac comb is equal to its own Fourier transform. This is significant because if Шаблон:Mvar is any Schwartz function, then the periodization of Шаблон:Mvar is given by the convolution <math display="block">(f * \operatorname{\text{Ш}})(x) = \sum_{n=-\infty}^\infty f(x-n).</math> In particular, <math display="block">(f*\operatorname{\text{Ш}})^\wedge = \widehat{f}\widehat{\operatorname{\text{Ш}}} = \widehat{f}\operatorname{\text{Ш}}</math> is precisely the Poisson summation formula.Шаблон:SfnШаблон:Sfn More generally, this formula remains to be true if Шаблон:Mvar is a tempered distribution of rapid descent or, equivalently, if <math>\widehat{f}</math> is a slowly growing, ordinary function within the space of tempered distributions.

Sokhotski–Plemelj theorem

The Sokhotski–Plemelj theorem, important in quantum mechanics, relates the delta function to the distribution Шаблон:Math, the Cauchy principal value of the function Шаблон:Math, defined by

<math display="block">\left\langle\operatorname{p.v.}\frac{1}{x}, \varphi\right\rangle = \lim_{\varepsilon\to 0^+}\int_{|x|>\varepsilon} \frac{\varphi(x)}{x}\,dx.</math>

Sokhotsky's formula states thatШаблон:Sfn

<math display="block">\lim_{\varepsilon\to 0^+} \frac{1}{x\pm i\varepsilon} = \operatorname{p.v.}\frac{1}{x} \mp i\pi\delta(x),</math>

Here the limit is understood in the distribution sense, that for all compactly supported smooth functions Шаблон:Mvar,

<math display="block">\int_{-\infty}^{\infty}\lim_{\varepsilon\to0^{+}}\frac{f(x)}{x\pm i\varepsilon}\,dx=\mp i\pi f(0)+\lim_{\varepsilon\to0^{+}}\int_{|x|>\varepsilon}\frac{f(x)}{x}\,dx.</math>

Relationship to the Kronecker delta

The Kronecker delta Шаблон:Mvar is the quantity defined by

<math display="block">\delta_{ij} = \begin{cases} 1 & i=j\\ 0 &i\not=j \end{cases} </math>

for all integers Шаблон:Mvar, Шаблон:Mvar. This function then satisfies the following analog of the sifting property: if Шаблон:Mvar (for Шаблон:Mvar in the set of all integers) is any doubly infinite sequence, then

<math display="block">\sum_{i=-\infty}^\infty a_i \delta_{ik}=a_k.</math>

Similarly, for any real or complex valued continuous function Шаблон:Mvar on Шаблон:Math, the Dirac delta satisfies the sifting property

<math display="block">\int_{-\infty}^\infty f(x)\delta(x-x_0)\,dx=f(x_0).</math>

This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function.Шаблон:Sfn

Applications

Probability theory

In probability theory and statistics, the Dirac delta function is often used to represent a discrete distribution, or a partially discrete, partially continuous distribution, using a probability density function (which is normally used to represent absolutely continuous distributions). For example, the probability density function Шаблон:Math of a discrete distribution consisting of points Шаблон:Math, with corresponding probabilities Шаблон:Math, can be written as

<math display="block">f(x) = \sum_{i=1}^n p_i \delta(x-x_i).</math>

As another example, consider a distribution in which 6/10 of the time returns a standard normal distribution, and 4/10 of the time returns exactly the value 3.5 (i.e. a partly continuous, partly discrete mixture distribution). The density function of this distribution can be written as

<math display="block">f(x) = 0.6 \, \frac {1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}} + 0.4 \, \delta(x-3.5).</math>

The delta function is also used to represent the resulting probability density function of a random variable that is transformed by continuously differentiable function. If Шаблон:Math is a continuous differentiable function, then the density of Шаблон:Mvar can be written as

<math display="block">f_Y(y) = \int_{-\infty}^{+\infty} f_X(x) \delta(y-g(x)) \,dx. </math>

The delta function is also used in a completely different way to represent the local time of a diffusion process (like Brownian motion). The local time of a stochastic process Шаблон:Math is given by <math display="block">\ell(x,t) = \int_0^t \delta(x-B(s))\,ds</math> and represents the amount of time that the process spends at the point Шаблон:Mvar in the range of the process. More precisely, in one dimension this integral can be written <math display="block">\ell(x,t) = \lim_{\varepsilon\to 0^+}\frac{1}{2\varepsilon}\int_0^t \mathbf{1}_{[x-\varepsilon,x+\varepsilon]}(B(s))\,ds</math> where <math>\mathbf{1}_{[x-\varepsilon,x+\varepsilon]}</math> is the indicator function of the interval <math>[x-\varepsilon,x+\varepsilon].</math>

Quantum mechanics

The delta function is expedient in quantum mechanics. The wave function of a particle gives the probability amplitude of finding a particle within a given region of space. Wave functions are assumed to be elements of the Hilbert space Шаблон:Math of square-integrable functions, and the total probability of finding a particle within a given interval is the integral of the magnitude of the wave function squared over the interval. A set Шаблон:Math of wave functions is orthonormal if they are normalized by

<math display="block">\langle\varphi_n \mid \varphi_m\rangle = \delta_{nm}</math>

where Шаблон:Mvar is the Kronecker delta. A set of orthonormal wave functions is complete in the space of square-integrable functions if any wave function Шаблон:Math can be expressed as a linear combination of the Шаблон:Math with complex coefficients:

<math display="block"> \psi = \sum c_n \varphi_n, </math>

with Шаблон:Math. Complete orthonormal systems of wave functions appear naturally as the eigenfunctions of the Hamiltonian (of a bound system) in quantum mechanics that measures the energy levels, which are called the eigenvalues. The set of eigenvalues, in this case, is known as the spectrum of the Hamiltonian. In bra–ket notation, as above, this equality implies the resolution of the identity:

<math display="block">I = \sum |\varphi_n\rangle\langle\varphi_n|.</math>

Here the eigenvalues are assumed to be discrete, but the set of eigenvalues of an observable may be continuous rather than discrete. An example is the position observable, Шаблон:Math. The spectrum of the position (in one dimension) is the entire real line and is called a continuous spectrum. However, unlike the Hamiltonian, the position operator lacks proper eigenfunctions. The conventional way to overcome this shortcoming is to widen the class of available functions by allowing distributions as well: that is, to replace the Hilbert space of quantum mechanics with an appropriate rigged Hilbert space.Шаблон:Sfn In this context, the position operator has a complete set of eigen-distributions, labeled by the points Шаблон:Mvar of the real line, given by

<math display="block">\varphi_y(x) = \delta(x-y).</math>

The eigenfunctions of position are denoted by Шаблон:Math in Dirac notation, and are known as position eigenstates.

Similar considerations apply to the eigenstates of the momentum operator, or indeed any other self-adjoint unbounded operator Шаблон:Mvar on the Hilbert space, provided the spectrum of Шаблон:Mvar is continuous and there are no degenerate eigenvalues. In that case, there is a set Шаблон:Math of real numbers (the spectrum), and a collection Шаблон:Mvar of distributions indexed by the elements of Шаблон:Math, such that

<math display="block">P\varphi_y = y\varphi_y.</math>

That is, Шаблон:Mvar are the eigenvectors of Шаблон:Mvar. If the eigenvectors are normalized so that

<math display="block">\langle \varphi_y,\varphi_{y'}\rangle = \delta(y-y')</math>

in the distribution sense, then for any test function Шаблон:Mvar,

<math display="block"> \psi(x) = \int_\Omega c(y) \varphi_y(x) \, dy</math>

where Шаблон:Math. That is, as in the discrete case, there is a resolution of the identity

<math display="block">I = \int_\Omega |\varphi_y\rangle\, \langle\varphi_y|\,dy</math>

where the operator-valued integral is again understood in the weak sense. If the spectrum of Шаблон:Mvar has both continuous and discrete parts, then the resolution of the identity involves a summation over the discrete spectrum Шаблон:Em an integral over the continuous spectrum.

The delta function also has many more specialized applications in quantum mechanics, such as the delta potential models for a single and double potential well.

Structural mechanics

The delta function can be used in structural mechanics to describe transient loads or point loads acting on structures. The governing equation of a simple mass–spring system excited by a sudden force impulse Шаблон:Mvar at time Шаблон:Math can be written

<math display="block">m \frac{d^2 \xi}{dt^2} + k \xi = I \delta(t),</math>

where Шаблон:Mvar is the mass, Шаблон:Mvar is the deflection, and Шаблон:Mvar is the spring constant.

As another example, the equation governing the static deflection of a slender beam is, according to Euler–Bernoulli theory,

<math display="block">EI \frac{d^4 w}{dx^4} = q(x),</math>

where Шаблон:Mvar is the bending stiffness of the beam, Шаблон:Mvar is the deflection, Шаблон:Mvar is the spatial coordinate, and Шаблон:Math is the load distribution. If a beam is loaded by a point force Шаблон:Mvar at Шаблон:Math, the load distribution is written

<math display="block">q(x) = F \delta(x-x_0).</math>

As the integration of the delta function results in the Heaviside step function, it follows that the static deflection of a slender beam subject to multiple point loads is described by a set of piecewise polynomials.

Also, a point moment acting on a beam can be described by delta functions. Consider two opposing point forces Шаблон:Mvar at a distance Шаблон:Mvar apart. They then produce a moment Шаблон:Math acting on the beam. Now, let the distance Шаблон:Mvar approach the limit zero, while Шаблон:Mvar is kept constant. The load distribution, assuming a clockwise moment acting at Шаблон:Math, is written

<math display="block">\begin{align} q(x) &= \lim_{d \to 0} \Big( F \delta(x) - F \delta(x-d) \Big) \\[4pt] &= \lim_{d \to 0} \left( \frac{M}{d} \delta(x) - \frac{M}{d} \delta(x-d) \right) \\[4pt] &= M \lim_{d \to 0} \frac{\delta(x) - \delta(x - d)}{d}\\[4pt] &= M \delta'(x). \end{align}</math>

Point moments can thus be represented by the derivative of the delta function. Integration of the beam equation again results in piecewise polynomial deflection.

See also

• Atom (measure theory)
• Laplacian of the indicator

References

• Шаблон:Citation.
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External links

• Шаблон:Commons category-inline
• Шаблон:Springer
• KhanAcademy.org video lesson
• The Dirac Delta function, a tutorial on the Dirac delta function.
• Video Lectures – Lecture 23, a lecture by Arthur Mattuck.
• The Dirac delta measure is a hyperfunction
• We show the existence of a unique solution and analyze a finite element approximation when the source term is a Dirac delta measure
• Non-Lebesgue measures on R. Lebesgue-Stieltjes measure, Dirac delta measure. Шаблон:Webarchive