Dirac delta function

 ( Fourier Analysis ) 

In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, 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. Thus it can be Heuristic as

such that

$$\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash delta(x)\; dx=1.$$

Since there is no function having this property, modelling 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.

To be specific, suppose that a billiard ball is at rest. At time $t=0$ it is struck by another ball, imparting it with a momentum , 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 ; the units of are s⁻¹.

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

Then the momentum at any time is found by integration:

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

Here the functions $F\_\{\backslash Delta\; t\}$ 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) $\backslash lim\_\{\backslash Delta\; t\backslash to\; 0^+\}F\_\{\backslash Delta\; t\}$ 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

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; F\_\{\backslash Delta\; t\}(t)\backslash ,dt\; =\; P,$$

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

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 . For example, the objects and are equal everywhere except at yet have integrals that are different. According to Lebesgue integration theory, if and are functions such that almost everywhere, then is integrable if and only if is integrable and the integrals of and are identical. A rigorous approach to regarding the Dirac delta function as a mathematical object in its own right uses measure theory or the theory of distributions.

Mathematicians refer to the same concept as a distribution rather than a function.

$$f(x)=\backslash frac\{1\}\{2\backslash pi\}\backslash int\_\{-\backslash infty\}^\backslash infty\backslash \; \backslash \; d\backslash alpha\; \backslash ,\; f(\backslash alpha)\; \backslash \; \backslash int\_\{-\backslash infty\}^\backslash infty\; dp\backslash \; \backslash cos\; (px-p\backslash alpha)\backslash \; ,$$

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

$$\backslash delta(x-\backslash alpha)=\backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_\{-\backslash infty\}^\backslash infty\; dp\backslash \; \backslash cos\; (px-p\backslash alpha)\; \backslash \; .$$

Later, Augustin Cauchy expressed the theorem using exponentials:

$$f(x)=\backslash frac\{1\}\{2\backslash pi\}\; \backslash int\_\{-\backslash infty\}\; ^\; \backslash infty\; \backslash \; e^\{ipx\}\backslash left(\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-ip\backslash alpha\; \}f(\backslash alpha)\backslash ,d\; \backslash alpha\; \backslash right)\; \backslash ,dp.$$

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

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 Norbert Wiener and Salomon Bochner works (around 1930) and culminating with the amalgamation into Laurent Schwartz theory of distributions (1945) ...",

and which is also constrained to satisfy the identity

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash delta(x)\; \backslash ,\; dx\; =\; 1.$$

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

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; f(x)\; \backslash ,\; \backslash delta(dx)\; =\; f(0)$$

for all continuous compactly supported functions . The measure 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

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; f(x)\backslash ,\; \backslash delta(x)\backslash ,\; dx\; =\; f(0)$$

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

As a probability measure on , the delta measure is characterized by its cumulative distribution function, which is the unit step function. See also for a different interpretation. Other conventions for the assigning the value of the Heaviside function at zero exist, and some of these are not consistent with what follows.

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

$$H(x)\; =\; \backslash int\_\{\backslash mathbf\{R\}\}\backslash mathbf\{1\}\_\{(-\backslash infty,x]\}(t)\backslash ,\backslash delta(dt)\; =\; \backslash delta\backslash !\backslash left((-\backslash infty,x]\backslash right),$$

the latter being the measure of this interval. Thus in particular the integration of the delta function against a continuous function can be properly understood as a Riemann–Stieltjes integral:

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; f(x)\backslash ,\backslash delta(dx)\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; f(x)\; \backslash ,dH(x).$$

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

A typical space of test functions consists of all on 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 by

for every test function .

For 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 on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer there is an integer and a constant such that for every test function , one has the inequality

$$\backslash left|S\backslash varphi\backslash right|\; \backslash le\; C\_N\; \backslash sum\_\{k=0\}^\{M\_N\}\backslash sup\_\{x\backslash in\; -N,N\}\; \backslash left|\backslash varphi^\{(k)\}(x)\backslash right|$$

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

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 , one has

$$\backslash delta\backslash varphi\; =\; -\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash varphi\text{'}(x)\backslash ,H(x)\backslash ,dx.$$

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

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash varphi(x)\backslash ,H\text{'}(x)\backslash ,dx\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash varphi(x)\backslash ,\backslash delta(x)\backslash ,dx,$$

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

$$-\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash varphi\text{'}(x)\backslash ,H(x)\backslash ,\; dx\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash varphi(x)\backslash ,dH(x).$$

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

$$\backslash int\_\{\backslash mathbf\{R\}^n\}\; f(\backslash mathbf\{x\})\backslash ,\backslash delta(d\backslash mathbf\{x\})\; =\; f(\backslash mathbf\{0\})$$

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

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

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

is the delta measure or unit mass concentrated at .

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 centered at the point is defined as the following distribution:

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

On a locally compact Hausdorff space , the Dirac delta measure concentrated at a point is the Radon measure associated with the Daniell integral () on compactly supported continuous functions .

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash delta(\backslash alpha\; x)\backslash ,dx\; =\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash delta(u)\backslash ,\backslash frac\{du\}$$

and so

Scaling property proof: $$\backslash int\backslash limits\_\{-\backslash infty\}^\{\backslash infty\}\; dx\backslash \; g(x)\; \backslash delta\; (ax)\; =\; \backslash frac\{1\}\{a\}\backslash int\backslash limits\_\{-\backslash infty\}^\{\backslash infty\}\; dx\text{'}\backslash \; g\backslash left(\backslash frac\{x\text{'}\}\{a\}\backslash right)\; \backslash delta\; (x\text{'})\; =\; \backslash frac\{1\}\{\; a\; \}g(0).$$ where a change of variable is used. If is negative, i.e., , then $$\backslash int\backslash limits\_\{-\backslash infty\}^\{\backslash infty\}\; dx\backslash \; g(x)\; \backslash delta\; (ax)\; =\; \backslash frac\{1\}\{-\backslash left\; \backslash vert\; a\; \backslash right\; \backslash vert\}\backslash int\backslash limits\_\{\backslash infty\}^\{-\backslash infty\}\; dx\text{'}\backslash \; g\backslash left(\backslash frac\{x\text{'}\}\{a\}\backslash right)\; \backslash delta\; (x\text{'})\; =\; \backslash frac\{1\}\{\backslash left\; \backslash vert\; a\; \backslash right\; \backslash vert\}\backslash int\backslash limits\_\{-\backslash infty\}^\{\backslash infty\}\; dx\text{'}\backslash \; g\backslash left(\backslash frac\{x\text{'}\}\{a\}\backslash right)\; \backslash delta\; (x\text{'})\; =\; \backslash frac\{1\}\{\backslash left\; \backslash vert\; a\; \backslash right\; \backslash vert\}g(0).$$ Thus,

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

$$\backslash delta(-x)\; =\; \backslash delta(x)$$

which is homogeneous of degree .

$$x\backslash ,\backslash delta(x)\; =\; 0.$$

More generally, $(x-a)^n\backslash delta(x-a)\; =0$ for all positive integers $n$.

Conversely, if , where and are distributions, then

$$f(x)\; =\; g(x)\; +c\; \backslash delta(x)$$

for some constant .

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; f(t)\; \backslash ,\backslash delta(t-T)\backslash ,dt\; =\; f(T).$$

This is sometimes referred to as the sifting property or the sampling property.

It follows that the effect of Convolution a function with the time-delayed Dirac delta is to time-delay by the same amount:

The sifting property holds under the precise condition that 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)

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash delta\; (\backslash xi-x)\; \backslash delta(x-\backslash eta)\; \backslash ,dx\; =\; \backslash delta(\backslash eta-\backslash xi).$$

$$\backslash int\_\{\backslash R\}\; \backslash delta\backslash bigl(g(x)\backslash bigr)\; f\backslash bigl(g(x)\backslash bigr)\; \backslash left|g\text{'}(x)\backslash right|\; dx\; =\; \backslash int\_\{g(\backslash R)\}\; \backslash delta(u)\backslash ,f(u)\backslash ,du$$

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

$$\backslash delta(g(x))\; =\; \backslash frac\{\backslash delta(x-x\_0)\}$$

It is natural therefore to the composition for continuously differentiable functions by

$$\backslash delta(g(x))\; =\; \backslash sum\_i\; \backslash frac\{\backslash delta(x-x\_i)\}$$

where the sum extends over all roots of , which are assumed to be simple root. Thus, for example

$$\backslash delta\backslash left(x^2-\backslash alpha^2\backslash right)\; =\; \backslash frac\{1\}\{2|\backslash alpha|\}\; \backslash Big\backslash delta\backslash left(x+\backslash alpha\backslash right)+\backslash delta\backslash left(x-\backslash alpha\backslash right)\backslash Big.$$

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

$$\backslash int\_\{-\backslash infty\}^\backslash infty\; f(x)\; \backslash ,\; \backslash delta(g(x))\; \backslash ,\; dx\; =\; \backslash sum\_\{i\}\backslash frac\{f(x\_i)\}$$

Under any reflection or rotation , the delta function is invariant, $$\backslash delta(\backslash rho\; \backslash boldsymbol\{x\})\; =\; \backslash delta(\backslash boldsymbol\{x\})~.$$

As in the one-variable case, it is possible to define the composition of with a bi-Lipschitz functionFurther refinement is possible, namely to submersions, although these require a more involved change of variables formula. uniquely so that the following holds $$\backslash int\_\{\backslash R^n\}\; \backslash delta(g(\backslash boldsymbol\{x\}))\backslash ,\; f(g(\backslash boldsymbol\{x\}))\backslash left|\backslash det\; g\text{'}(\backslash boldsymbol\{x\})\backslash right|\; d\backslash boldsymbol\{x\}\; =\; \backslash int\_\{g(\backslash R^n)\}\; \backslash delta(\backslash boldsymbol\{u\})\; f(\backslash boldsymbol\{u\})\backslash ,d\backslash boldsymbol\{u\}$$ for all compactly supported functions .

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 such that the gradient of is nowhere zero, the following identity holds $$\backslash int\_\{\backslash R^n\}\; f(\backslash boldsymbol\{x\})\; \backslash ,\; \backslash delta(g(\backslash boldsymbol\{x\}))\; \backslash ,d\backslash boldsymbol\{x\}\; =\; \backslash int\_\{g^\{-1\}(0)\}\backslash frac\{f(\backslash boldsymbol\{x\})\}$$

More generally, if is a smooth hypersurface of , then we can associate to the distribution that integrates any compactly supported smooth function over : $$\backslash delta\_Sg\; =\; \backslash int\_S\; g(\backslash boldsymbol\{s\})\backslash ,d\backslash sigma(\backslash boldsymbol\{s\})$$

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

$$-\backslash int\_\{\backslash R^n\}g(\backslash boldsymbol\{x\})\backslash ,\backslash frac\{\backslash partial\; 1\_D(\backslash boldsymbol\{x\})\}\{\backslash partial\; n\}\backslash ,d\backslash boldsymbol\{x\}=\backslash int\_S\backslash ,g(\backslash boldsymbol\{s\})\backslash ,\; d\backslash sigma(\backslash boldsymbol\{s\}),$$

where is the outward normal.

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

$$\backslash delta(\backslash boldsymbol\{r\}-\backslash boldsymbol\{r\}\_0)\; =\; \backslash begin\{cases\}$$

   \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
     

The first equality here is a kind of integration by parts, for if were a true function then $$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash delta\text{'}(x)\backslash varphi(x)\backslash ,dx\; =\; \backslash delta(x)\backslash varphi(x)|\_\{-\backslash infty\}^\{\backslash infty\}\; -\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash delta(x)\; \backslash varphi\text{'}(x)\backslash ,dx\; =\; -\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash delta(x)\; \backslash varphi\text{'}(x)\backslash ,dx\; =\; -\backslash varphi\text{'}(0).$$

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

$$\backslash delta^\{(k)\}\backslash varphi\; =\; (-1)^k\; \backslash varphi^\{(k)\}(0).$$

In particular, is an infinitely differentiable distribution.

The first derivative of the delta function is the distributional limit of the difference quotients: $$\backslash delta\text{'}(x)\; =\; \backslash lim\_\{h\backslash to\; 0\}\; \backslash frac\{\backslash delta(x+h)-\backslash delta(x)\}\{h\}.$$

More properly, one has $$\backslash delta\text{'}\; =\; \backslash lim\_\{h\backslash to\; 0\}\; \backslash frac\{1\}\{h\}(\backslash tau\_h\backslash delta\; -\; \backslash delta)$$ where is the translation operator, defined on functions by , and on a distribution by $$(\backslash tau\_h\; S)\backslash varphi\; =\; S\backslash tau\_\{-h\}\backslash varphi.$$

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 unit doublet.

The derivative of the delta function satisfies a number of basic properties, including: $$\backslash begin\{align\}$$

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

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

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

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

$$\backslash delta\text{'}*f\; =\; \backslash delta*f\text{'}\; =\; f\text{'},$$

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

$$\backslash left\backslash langle\; \backslash partial^\backslash alpha\; \backslash delta\_\{a\},\; \backslash ,\; \backslash varphi\; \backslash right\backslash rangle\; =\; (-1)^$$

That is, the -th derivative of is the distribution whose value on any test function is the -th derivative of at (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 .

Higher derivatives enter into mathematics naturally as the building blocks for the complete structure of distributions with point support. If is any distribution on supported on the set consisting of a single point, then there is an integer and coefficients such that $$S\; =\; \backslash sum\_$$

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.

where the operator is rigorously defined as the Fourier multiplier $$\backslash mathcal\{F\}\backslash left\backslash left(-\backslash frac\{\backslash partial^2\}\{\backslash partial(\backslash xi)\; =\; |2\backslash pi\backslash xi|\backslash mathcal\{F\}f(\backslash xi).$$

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 :

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) $$\backslash eta\_\backslash varepsilon(x)=\backslash frac\{1\}\{\backslash pi\; x\}\backslash sin\backslash left(\backslash frac\{x\}\{\backslash varepsilon\}\backslash right)=\backslash frac\{1\}\{2\backslash pi\}\backslash int\_\{-\backslash frac\{1\}\{\backslash varepsilon\}\}^\{\backslash frac\{1\}\{\backslash varepsilon\}\}\; \backslash cos(kx)\backslash ,dk$$

and the Bessel function $$\backslash eta\_\backslash varepsilon(x)\; =\; \backslash frac\{1\}\{\backslash varepsilon\}J\_\{\backslash frac\{1\}\{\backslash varepsilon\}\}\; \backslash left(\backslash frac\{x+1\}\{\backslash varepsilon\}\backslash right).$$

where is a differential operator on , is to seek first a fundamental solution, which is a solution of the equation $$Lu=\backslash delta.$$

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 $$Lu=h$$

where is a plane wave function, meaning that it has the form $$h\; =\; h(x\backslash cdot\backslash xi)$$

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 $$g(s)\; =\; \backslash operatorname\{Re\}\backslash left\backslash frac\{-s^k\backslash log(-is)\}\{k!(2\backslash pi\; =\backslash begin\{cases\}\; \backslash frac$$

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 : $$\backslash delta(x)\; =\; \backslash Delta\_x^\{(n+k)/2\}\; \backslash int\_\{S^\{n-1\}\}\; g(x\backslash cdot\backslash xi)\backslash ,d\backslash omega\_\backslash xi.$$

The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function , $$\backslash varphi(x)\; =\; \backslash int\_\{\backslash mathbf\{R\}^n\}\backslash varphi(y)\backslash ,dy\backslash ,\backslash Delta\_x^\{\backslash frac\{n+k\}\{2\}\}\; \backslash int\_\{S^\{n-1\}\}\; g((x-y)\backslash cdot\backslash xi)\backslash ,d\backslash omega\_\backslash xi.$$

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

where is the Radon transform of : $$R\backslash varphi(\backslash xi,p)\; =\; \backslash int\_\{x\backslash cdot\backslash xi=p\}\; f(x)\backslash ,d^\{n-1\}x.$$

An alternative equivalent expression of the plane wave decomposition is: $$\backslash delta(x)\; =\; \backslash begin\{cases\}$$

 \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}
     

$$\backslash widehat\{\backslash delta\}(\backslash xi)=\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-2\backslash pi\; i\; x\; \backslash xi\}\; \backslash ,\backslash delta(x)dx\; =\; 1.$$

Properly speaking, the Fourier transform of a distribution is defined by imposing of the Fourier transform under the duality pairing $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ of tempered distributions with Schwartz functions. Thus $\backslash widehat\{\backslash delta\}$ is defined as the unique tempered distribution satisfying

$$\backslash langle\backslash widehat\{\backslash delta\},\backslash varphi\backslash rangle\; =\; \backslash langle\backslash delta,\backslash widehat\{\backslash varphi\}\backslash rangle$$

for all Schwartz functions . And indeed it follows from this that $\backslash widehat\{\backslash delta\}=1.$

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

$$S*\backslash delta\; =\; S.$$

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 $$\backslash int\_\{-\backslash infty\}^\backslash infty\; 1\; \backslash cdot\; e^\{2\backslash pi\; i\; x\backslash xi\}\backslash ,d\backslash xi\; =\; \backslash delta(x)$$ and more rigorously, it follows since $$\backslash langle\; 1,\; \backslash widehat\{f\}\backslash rangle\; =\; f(0)\; =\; \backslash langle\backslash delta,f\backslash rangle$$ 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 $$\backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{i\; 2\backslash pi\; \backslash xi\_1\; t\}\; \backslash lefte^\{i^*\backslash ,dt\; =\; \backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-i\; 2\backslash pi\; (\backslash xi\_2\; -\; \backslash xi\_1)\; t\}\; \backslash ,dt\; =\; \backslash delta(\backslash xi\_2\; -\; \backslash xi\_1).$$

This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution $$f(t)\; =\; e^\{i2\backslash pi\backslash xi\_1\; t\}$$ is $$\backslash widehat\{f\}(\backslash xi\_2)\; =\; \backslash delta(\backslash xi\_1-\backslash xi\_2)$$ 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 $$\backslash int\_\{0\}^\{\backslash infty\}\backslash delta(t-a)\backslash ,e^\{-st\}\; \backslash ,\; dt=e^\{-sa\}.$$

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

$$F\_N(x)\; =\; \backslash frac1N\backslash sum\_\{n=0\}^\{N-1\}\; D\_n(x)\; =\; \backslash frac\{1\}\{N\}\backslash left(\backslash frac\{\backslash sin\; \backslash frac\{Nx\}\{2\}\}\{\backslash sin\; \backslash frac\{x\}\{2\}\}\backslash right)^2.$$

The Fejér kernels tend to the delta function in a stronger sense thatIn the terminology of , the Fejér kernel is a Dirac sequence, whereas the Dirichlet kernel is not.

$$\backslash int\_\{-\backslash pi\}^\{\backslash pi\}\; F\_N(x)f(x)\backslash ,dx\; \backslash to\; 2\backslash pi\; f(0)$$

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.

is automatically continuous, and satisfies in particular

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 .

$$f(z)\; =\; \backslash frac\{1\}\{2\backslash pi\; i\}\; \backslash oint\_\{\backslash partial\; D\}\; \backslash frac\{f(\backslash zeta)\backslash ,d\backslash zeta\}\{\backslash zeta-z\},\backslash quad\; z\backslash in\; D$$

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:

$$\backslash delta\_zf\; =\; f(z)\; =\; \backslash frac\{1\}\{2\backslash pi\; i\}\; \backslash oint\_\{\backslash partial\; D\}\; \backslash frac\{f(\backslash zeta)\backslash ,d\backslash zeta\}\{\backslash zeta-z\}.$$

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 $H(D)\backslash cap\; L^2(D)$ of square-integrable holomorphic functions in an open set $D\backslash subset\backslash mathbb\; C^n$. 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 $H(D)\backslash cap\; L^2(D)$ at a point $z$ of $D$ is a continuous functional, and so by the Riesz representation theorem, is represented by integration against a kernel $K\_z(\backslash zeta)$, 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

| year = 2011
| publisher = Elsevier
| isbn = 978-0-08-054996-5
| language = en