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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.

The Dirac delta function as the limit (in the sense of distributions) of the sequence of zero-centered normal distributions δa(x)=1|a|πe−(x/a)2{displaystyle delta _{a}(x)={frac {1}{left|aright|{sqrt {pi }}}}mathrm {e} ^{-(x/a)^{2}}} as a→0{displaystyle arightarrow 0}.

In mathematics, the Dirac delta function (δ function) is a generalized function or distribution introduced by the physicist Paul Dirac. It is used to model the density of an idealized point mass or point charge as a function equal to zero everywhere except for zero and whose integral over the entire real line is equal to one.^[1][2][3] As there is no function that has these properties, the computations made by the theoretical physicists appeared to mathematicians as nonsense until the introduction of distributions by Laurent Schwartz to formalize and validate the computations. As a distribution, the Dirac delta function is a linear functional that maps every function to its value at zero.^[4][5] The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is a discrete analog of the Dirac delta function.

In engineering and signal processing, the delta function, also known as the unit impulse symbol,^[6] may be regarded through its Laplace transform, as coming from the boundary values of a complex analytic function of a complex variable. The formal rules obeyed by this function are part of the operational calculus, a standard tool kit of physics and engineering. In many applications, the Dirac delta is regarded as a kind of limit (a weak limit) of a sequence of functions having a tall spike at the origin (in theory of distributions, this is a true limit). The approximating functions of the sequence are thus "approximate" or "nascent" delta functions.

Motivation and overview

The graph of the delta function is usually thought of as following the whole x-axis and the positive y-axis. 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 delta function. 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 t=0{displaystyle t=0} it is struck by another ball, imparting it with a momentum P, in kg m/s{displaystyle {text{kg m}}/{text{s}}}. 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 Pδ(t){displaystyle Pdelta (t)}. (The units of δ(t){displaystyle delta (t)} are s−1{displaystyle s^{-1}}.)

To model this situation more rigorously, suppose that the force instead is uniformly distributed over a small time interval Δt{displaystyle Delta t}. That is,

FΔt(t)={P/Δt0<t<Δt,0otherwise.{displaystyle F_{Delta t}(t)={begin{cases}P/Delta t&0<t<Delta t,\0&{text{otherwise}}.end{cases}}}

Then the momentum at any time t is found by integration:

p(t)=∫0tFΔt(τ)dτ={Pt>ΔtPt/Δt0<t<Δt0otherwise.{displaystyle p(t)=int _{0}^{t}F_{Delta t}(tau ),dtau ={begin{cases}P&t>Delta t\Pt/Delta t&0<t<Delta t\0&{text{otherwise.}}end{cases}}}

Now, the model situation of an instantaneous transfer of momentum requires taking the limit as Δt→0{displaystyle Delta tto 0}, giving

p(t)={Pt>00t≤0.{displaystyle p(t)={begin{cases}P&t>0\0&tleq 0.end{cases}}}

Here the functions FΔt{displaystyle F_{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 ordinary calculus) limΔt→0FΔt{displaystyle lim _{Delta tto 0}F_{Delta t}} is zero everywhere but a single point, where it is infinite. To make proper sense of the delta function, we should instead insist that the property

∫−∞∞FΔt(t)dt=P,{displaystyle int _{-infty }^{infty }F_{Delta t}(t),dt=P,}

which holds for all Δt>0{displaystyle Delta t>0}, should continue to hold in the limit. So, in the equation F(t)=Pδ(t)=limΔt→0FΔt(t){displaystyle F(t)=Pdelta (t)=lim _{Delta tto 0}F_{Delta t}(t)}, it is understood that the limit is always taken outside the integral.

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.

Despite its name, the delta function is not truly a function, at least not a usual one with range in real numbers. For example, the objects f(x) = δ(x) and g(x) = 0 are equal everywhere except at x = 0 yet have integrals that are different. According to Lebesgue integration theory, if f and g are functions such that f = g almost everywhere, then f is integrable if and only if g is integrable and the integrals of f and g 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:^[7]

f(x)=12π∫−∞∞  dαf(α) ∫−∞∞dp cos⁡(px−pα) ,{displaystyle f(x)={frac {1}{2pi }}int _{-infty }^{infty } dalpha ,f(alpha ) int _{-infty }^{infty }dp cos(px-palpha ) ,}

which is tantamount to the introduction of the δ-function in the form:^[8]

δ(x−α)=12π∫−∞∞dp cos⁡(px−pα) .{displaystyle delta (x-alpha )={frac {1}{2pi }}int _{-infty }^{infty }dp cos(px-palpha ) .}

Later, Augustin Cauchy expressed the theorem using exponentials:^[9][10]

f(x)=12π∫−∞∞ eipx(∫−∞∞e−ipαf(α) dα) dp.{displaystyle f(x)={frac {1}{2pi }}int _{-infty }^{infty } e^{ipx}left(int _{-infty }^{infty }e^{-ipalpha }f(alpha ) dalpha right) dp.}

Cauchy pointed out that in some circumstances the order of integration in this result is significant (contrast Fubini's theorem).^[11][12]

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:

f(x)=12π∫−∞∞eipx(∫−∞∞e−ipαf(α) dα) dp=12π∫−∞∞(∫−∞∞eipxe−ipα dp)f(α) dα=∫−∞∞δ(x−α)f(α) dα,{displaystyle {begin{aligned}f(x)&={frac {1}{2pi }}int _{-infty }^{infty }e^{ipx}left(int _{-infty }^{infty }e^{-ipalpha }f(alpha ) dalpha right) dp\[4pt]&={frac {1}{2pi }}int _{-infty }^{infty }left(int _{-infty }^{infty }e^{ipx}e^{-ipalpha } dpright)f(alpha ) dalpha =int _{-infty }^{infty }delta (x-alpha )f(alpha ) dalpha ,end{aligned}}}

where the δ-function is expressed as:

δ(x−α)=12π∫−∞∞eip(x−α) dp .{displaystyle delta (x-alpha )={frac {1}{2pi }}int _{-infty }^{infty }e^{ip(x-alpha )} dp .}

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:^[13]

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) in order 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) ...",^[14] 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.^[15]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.^[16] The Dirac delta function as such was introduced as a "convenient notation" by Paul Dirac in his influential 1930 book The Principles of Quantum Mechanics.^[17] He called it the "delta function" since he used it as a continuous analogue of the discrete Kronecker delta.

Definitions

The Dirac delta can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,

δ(x)={+∞,x=00,x≠0{displaystyle delta (x)={begin{cases}+infty ,&x=0\0,&xneq 0end{cases}}}

and which is also constrained to satisfy the identity

∫−∞∞δ(x)dx=1.{displaystyle int _{-infty }^{infty }delta (x),dx=1.}^[18]

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.^[17] The Dirac delta function can be rigorously defined either as a distribution or as a measure.

As a measure

One way to rigorously capture the notion of the Dirac delta function is to define a measure, which accepts a subset A of the real line R as an argument, and returns δ(A) = 1 if 0 ∈ A, and δ(A) = 0 otherwise.^[19] If the delta function is conceptualized as modeling an idealized point mass at 0, then δ(A) represents the mass contained in the set A. One may then define the integral against δ 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 δ satisfies

∫−∞∞f(x)δ{dx}=f(0){displaystyle int _{-infty }^{infty }f(x),delta {dx}=f(0)}

for all continuous compactly supported functions f. The measure δ 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 — no true function for which the property

∫−∞∞f(x)δ(x)dx=f(0){displaystyle int _{-infty }^{infty }f(x)delta (x),dx=f(0)}

holds.^[20] 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 R, the delta measure is characterized by its cumulative distribution function, which is the unit step function^[21]

H(x)={1if x≥00if x<0.{displaystyle H(x)={begin{cases}1&{text{if }}xgeq 0\0&{text{if }}x<0.end{cases}}}

This means that H(x) is the integral of the cumulative indicator function 1_{(−∞, x]} with respect to the measure δ; to wit,

H(x)=∫R1(−∞,x](t)δ{dt}=δ(−∞,x].{displaystyle H(x)=int _{mathbf {R} }mathbf {1} _{(-infty ,x]}(t),delta {dt}=delta (-infty ,x].}

Thus in particular the integral of the delta function against a continuous function can be properly understood as a Riemann–Stieltjes integral:^[22]

∫−∞∞f(x)δ{dx}=∫−∞∞f(x)dH(x).{displaystyle int _{-infty }^{infty }f(x)delta {dx}=int _{-infty }^{infty }f(x),dH(x).}

All higher moments of δ 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 in relation to how it affects other functions when "integrated" against them. 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 φ. 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 R 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^[23]

δ[φ]=φ(0){displaystyle delta [varphi ]=varphi (0)}

(1)

for every test function φ.

For δ 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 S on the space of test functions to define a distribution, it is necessary and sufficient that, for every positive integer N there is an integer M_N and a constant C_N such that for every test function φ, one has the inequality^[24]

|S[φ]|≤CN∑k=0MNsupx∈[−N,N]|φ(k)(x)|.{displaystyle |S[varphi ]|leq C_{N}sum _{k=0}^{M_{N}}sup _{xin [-N,N]}|varphi ^{(k)}(x)|.}

With the δ distribution, one has such an inequality (with C_N = 1) with M_N = 0 for all N. Thus δ is a distribution of order zero. It is, furthermore, a distribution with compact support (the support being {0}).

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

δ[φ]=−∫−∞∞φ′(x)H(x)dx.{displaystyle delta [varphi ]=-int _{-infty }^{infty }varphi '(x)H(x),dx.}

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

∫−∞∞φ(x)H′(x)dx=∫−∞∞φ(x)δ(x)dx,{displaystyle int _{-infty }^{infty }varphi (x)H'(x),dx=int _{-infty }^{infty }varphi (x)delta (x),dx,}

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

−∫−∞∞φ′(x)H(x)dx=∫−∞∞φ(x)dH(x).{displaystyle -int _{-infty }^{infty }varphi '(x)H(x),dx=int _{-infty }^{infty }varphi (x),dH(x).}

In the context of measure theory, the Dirac measure gives rise to a distribution by integration. Conversely, equation (1) defines a Daniell integral on the space of all compactly supported continuous functions φ which, by the Riesz representation theorem, can be represented as the Lebesgue integral of φ 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 n-dimensional Euclidean space Rⁿ as the measure such that

∫Rnf(x)δ{dx}=f(0){displaystyle int _{mathbf {R} ^{n}}f(mathbf {x} )delta {dmathbf {x} }=f(mathbf {0} )}

for every compactly supported continuous function f. As a measure, the n-dimensional delta function is the product measure of the 1-dimensional delta functions in each variable separately. Thus, formally, with x = (x₁, x₂, ..., x_n), one has^[6]

δ(x)=δ(x1)δ(x2)⋯δ(xn).{displaystyle delta (mathbf {x} )=delta (x_{1})delta (x_{2})cdots delta (x_{n}).}

(2)

The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case.^[25] However, despite widespread use in engineering contexts, (2) should be manipulated with care, since the product of distributions can only be defined under quite narrow circumstances.^[26]

The notion of a Dirac measure makes sense on any set.^[19] Thus if X is a set, x₀ ∈ X is a marked point, and Σ is any sigma algebra of subsets of X, then the measure defined on sets A ∈ Σ by

δx0(A)={1if x0∈A0if x0∉A{displaystyle delta _{x_{0}}(A)={begin{cases}1&{text{if }}x_{0}in A\0&{text{if }}x_{0}notin Aend{cases}}}

is the delta measure or unit mass concentrated at x₀.

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 M centered at the point x₀ ∈ M is defined as the following distribution:

δx0[φ]=φ(x0){displaystyle delta _{x_{0}}[varphi ]=varphi (x_{0})}

(3)

for all compactly supported smooth real-valued functions φ on M.^[27] A common special case of this construction is when M is an open set in the Euclidean space Rⁿ.

On a locally compact Hausdorff space X, the Dirac delta measure concentrated at a point x is the Radon measure associated with the Daniell integral (3) on compactly supported continuous functions φ.^[28] 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 x0↦δx0{displaystyle x_{0}mapsto delta _{x_{0}}} is a continuous embedding of X into the space of finite Radon measures on X, equipped with its vague topology. Moreover, the convex hull of the image of X under this embedding is dense in the space of probability measures on X.^[29]

Properties

Scaling and symmetry

The delta function satisfies the following scaling property for a non-zero scalar α:^[30]

∫−∞∞δ(αx)dx=∫−∞∞δ(u)du|α|=1|α|{displaystyle int _{-infty }^{infty }delta (alpha x),dx=int _{-infty }^{infty }delta (u),{frac {du}{|alpha |}}={frac {1}{|alpha |}}}

and so

δ(αx)=δ(x)|α|.{displaystyle delta (alpha x)={frac {delta (x)}{|alpha |}}.}

(4)

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

δ(−x)=δ(x){displaystyle delta (-x)=delta (x)}

which is homogeneous of degree −1.

Algebraic properties

The distributional product of δ with x is equal to zero:

xδ(x)=0.{displaystyle xdelta (x)=0.}

Conversely, if xf(x) = xg(x), where f and g are distributions, then

f(x)=g(x)+cδ(x){displaystyle f(x)=g(x)+cdelta (x)}

for some constant c.^[31]

Translation

The integral of the time-delayed Dirac delta is given by:

∫−∞∞f(t)δ(t−T)dt=f(T).{displaystyle int _{-infty }^{infty }f(t)delta (t-T),dt=f(T).}

This is sometimes referred to as the sifting property^[32] or the sampling property. The delta function is said to "sift out" the value at t = T.

It follows that the effect of convolving a function f(t) with the time-delayed Dirac delta is to time-delay f(t) by the same amount:

(f(t)∗δ(t−T)){displaystyle (f(t)*delta (t-T))}	=def ∫−∞∞f(τ)δ(t−T−τ)dτ{displaystyle {stackrel {mathrm {def} }{=}} int _{-infty }^{infty }f(tau )delta (t-T-tau ),dtau }
	=∫−∞∞f(τ)δ(τ−(t−T))dτ{displaystyle =int limits _{-infty }^{infty }f(tau )delta (tau -(t-T)),dtau }       (using  (4): δ(−x)=δ(x){displaystyle delta (-x)=delta (x)})
	=f(t−T).{displaystyle =f(t-T).}

This holds under the precise condition that f 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)

∫−∞∞δ(ξ−x)δ(x−η)dx=δ(ξ−η).{displaystyle int _{-infty }^{infty }delta (xi -x)delta (x-eta ),dx=delta (xi -eta ).}

Composition with a function

More generally, the delta distribution may be composed with a smooth function g(x) in such a way that the familiar change of variables formula holds, that

∫Rδ(g(x))f(g(x))|g′(x)|dx=∫g(R)δ(u)f(u)du{displaystyle int _{mathbf {R} }delta {bigl (}g(x){bigr )}f{bigl (}g(x){bigr )}|g'(x)|,dx=int _{g(mathbf {R} )}delta (u)f(u),du}

provided that g is a continuously differentiable function with g′ nowhere zero.^[33] That is, there is a unique way to assign meaning to the distribution δ∘g{displaystyle delta circ g} so that this identity holds for all compactly supported test functions f. Therefore, the domain must be broken up to exclude the g′ = 0 point. This distribution satisfies δ(g(x)) = 0 if g is nowhere zero, and otherwise if g has a real root at x₀, then

δ(g(x))=δ(x−x0)|g′(x0)|.{displaystyle delta (g(x))={frac {delta (x-x_{0})}{|g'(x_{0})|}}.}

It is natural therefore to define the composition δ(g(x)) for continuously differentiable functions g by

δ(g(x))=∑iδ(x−xi)|g′(xi)|{displaystyle delta (g(x))=sum _{i}{frac {delta (x-x_{i})}{|g'(x_{i})|}}}

where the sum extends over all roots of g(x), which are assumed to be simple.^[33] Thus, for example

δ(x2−α2)=12|α|[δ(x+α)+δ(x−α)].{displaystyle delta left(x^{2}-alpha ^{2}right)={frac {1}{2|alpha |}}{Big [}delta left(x+alpha right)+delta left(x-alpha right){Big ]}.}

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

∫−∞∞f(x)δ(g(x))dx=∑if(xi)|g′(xi)|.{displaystyle int _{-infty }^{infty }f(x),delta (g(x)),dx=sum _{i}{frac {f(x_{i})}{|g'(x_{i})|}}.}

Properties in n dimensions

The delta distribution in an n-dimensional space satisfies the following scaling property instead,

δ(αx)=|α|−nδ(x) ,{displaystyle delta (alpha mathbf {x} )=|alpha |^{-n}delta (mathbf {x} )~,}

so that δ is a homogeneous distribution of degree −n.

Under any reflection or rotation ρ, the delta function is invariant,

δ(ρx)=δ(x) .{displaystyle delta (rho mathbf {x} )=delta (mathbf {x} )~.}

As in the one-variable case, it is possible to define the composition of δ with a bi-Lipschitz function^[34]g: Rⁿ → Rⁿ uniquely so that the identity

∫Rnδ(g(x))f(g(x))|detg′(x)|dx=∫g(Rn)δ(u)f(u)du{displaystyle int _{mathbf {R} ^{n}}delta (g(mathbf {x} )),f(g(mathbf {x} ))left|det g'(mathbf {x} )right|,dmathbf {x} =int _{g(mathbf {R} ^{n})}delta (mathbf {u} )f(mathbf {u} ),dmathbf {u} }

for all compactly supported functions f.

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 g: Rⁿ → R such that the gradient of g is nowhere zero, the following identity holds^[35]

∫Rnf(x)δ(g(x))dx=∫g−1(0)f(x)|∇g|dσ(x){displaystyle int _{mathbf {R} ^{n}}f(mathbf {x} ),delta (g(mathbf {x} )),dmathbf {x} =int _{g^{-1}(0)}{frac {f(mathbf {x} )}{|mathbf {nabla } g|}},dsigma (mathbf {x} )}

where the integral on the right is over g⁻¹(0), the (n − 1)-dimensional surface defined by g(x) = 0 with respect to the Minkowski content measure. This is known as a simple layer integral.

More generally, if S is a smooth hypersurface of Rⁿ, then we can associate to S the distribution that integrates any compactly supported smooth function g over S:

δS[g]=∫Sg(s)dσ(s){displaystyle delta _{S}[g]=int _{S}g(mathbf {s} ),dsigma (mathbf {s} )}

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

−∫Rng(x)∂1D(x)∂ndx=∫Sg(s)dσ(s),{displaystyle -int _{mathbf {R} ^{n}}g(mathbf {x} ),{frac {partial 1_{D}(mathbf {x} )}{partial n}};dmathbf {x} =int _{S},g(mathbf {s} );dsigma (mathbf {s} ),}

where n is the outward normal.^[36][37] For a proof, see e.g. the article on the surface delta function.

Fourier transform

The delta function is a tempered distribution, and therefore it has a well-defined Fourier transform. Formally, one finds^[38]

δ^(ξ)=∫−∞∞e−2πixξδ(x)dx=1.{displaystyle {widehat {delta }}(xi )=int _{-infty }^{infty }e^{-2pi ixxi }delta (x),dx=1.}

Properly speaking, the Fourier transform of a distribution is defined by imposing self-adjointness of the Fourier transform under the duality pairing ⟨⋅,⋅⟩{displaystyle langle cdot ,cdot rangle } of tempered distributions with Schwartz functions. Thus δ^{displaystyle {widehat {delta }}} is defined as the unique tempered distribution satisfying

⟨δ^,φ⟩=⟨δ,φ^⟩{displaystyle langle {widehat {delta }},varphi rangle =langle delta ,{widehat {varphi }}rangle }

for all Schwartz functions φ. And indeed it follows from this that δ^=1.{displaystyle {widehat {delta }}=1.}

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

S∗δ=S.{displaystyle S*delta =S.}

That is to say that δ 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 δ, and once it is known, it characterizes the system completely. See LTI system theory § Impulse response and convolution.

The inverse Fourier transform of the tempered distribution f(ξ) = 1 is the delta function. Formally, this is expressed

∫−∞∞1⋅e2πixξdξ=δ(x){displaystyle int _{-infty }^{infty }1cdot e^{2pi ixxi },dxi =delta (x)}

and more rigorously, it follows since

⟨1,f∨⟩=f(0)=⟨δ,f⟩{displaystyle langle 1,f^{vee }rangle =f(0)=langle delta ,frangle }

for all Schwartz functions f.

In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on R. Formally, one has

∫−∞∞ei2πξ1t[ei2πξ2t]∗dt=∫−∞∞e−i2π(ξ2−ξ1)tdt=δ(ξ2−ξ1).{displaystyle int _{-infty }^{infty }e^{i2pi xi _{1}t}left[e^{i2pi xi _{2}t}right]^{*},dt=int _{-infty }^{infty }e^{-i2pi (xi _{2}-xi _{1})t},dt=delta (xi _{2}-xi _{1}).}

This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution

f(t)=ei2πξ1t{displaystyle f(t)=e^{i2pi xi _{1}t}}

is

f^(ξ2)=δ(ξ1−ξ2){displaystyle {widehat {f}}(xi _{2})=delta (xi _{1}-xi _{2})}

which again follows by imposing self-adjointness of the Fourier transform.

By analytic continuation of the Fourier transform, the Laplace transform of the delta function is found to be^[39]

∫0∞δ(t−a)e−stdt=e−sa.{displaystyle int _{0}^{infty }delta (t-a)e^{-st},dt=e^{-sa}.}

Distributional derivatives

The distributional derivative of the Dirac delta distribution is the distribution δ′ defined on compactly supported smooth test functions φ by^[40]

δ′[φ]=−δ[φ′]=−φ′(0).{displaystyle delta '[varphi ]=-delta [varphi ']=-varphi '(0).}

The first equality here is a kind of integration by parts, for if δ were a true function then

∫−∞∞δ′(x)φ(x)dx=−∫−∞∞δ(x)φ′(x)dx.{displaystyle int _{-infty }^{infty }delta '(x)varphi (x),dx=-int _{-infty }^{infty }delta (x)varphi '(x),dx.}

The k-th derivative of δ is defined similarly as the distribution given on test functions by

δ(k)[φ]=(−1)kφ(k)(0).{displaystyle delta ^{(k)}[varphi ]=(-1)^{k}varphi ^{(k)}(0).}

In particular, δ is an infinitely differentiable distribution.

The first derivative of the delta function is the distributional limit of the difference quotients:^[41]

δ′(x)=limh→0δ(x+h)−δ(x)h.{displaystyle delta '(x)=lim _{hto 0}{frac {delta (x+h)-delta (x)}{h}}.}

More properly, one has

δ′=limh→01h(τhδ−δ){displaystyle delta '=lim _{hto 0}{frac {1}{h}}(tau _{h}delta -delta )}

where τ_h is the translation operator, defined on functions by τ_hφ(x) = φ(x + h), and on a distribution S by

(τhS)[φ]=S[τ−hφ].{displaystyle (tau _{h}S)[varphi ]=S[tau _{-h}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 doublet function.^[42]

The derivative of the delta function satisfies a number of basic properties, including:

ddxδ(−x)=ddxδ(x)δ′(−x)=−δ′(x)xδ′(x)=−δ(x).{displaystyle {begin{aligned}&{frac {d}{dx}}delta (-x)={frac {d}{dx}}delta (x)\[8pt]&delta '(-x)=-delta '(x)\[8pt]&xdelta '(x)=-delta (x).end{aligned}}}^[43]

Furthermore, the convolution of δ′ with a compactly supported smooth function f is

δ′∗f=δ∗f′=f′,{displaystyle delta '*f=delta *f'=f',}

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

Higher dimensions

More generally, on an open set U in the n-dimensional Euclidean space Rⁿ, the Dirac delta distribution centered at a point a ∈ U is defined by^[44]

δa[φ]=φ(a){displaystyle delta _{a}[varphi ]=varphi (a)}

for all φ ∈ S(U), the space of all smooth compactly supported functions on U. If α = (α₁, ..., α_n) is any multi-index and ∂^α denotes the associated mixed partial derivative operator, then the αth derivative ∂^αδ_a of δ_a is given by^[44]

⟨∂αδa,φ⟩=(−1)|α|⟨δa,∂αφ⟩=(−1)|α|∂αφ(x)|x=a for all φ∈S(U).{displaystyle leftlangle partial ^{alpha }delta _{a},varphi rightrangle =(-1)^{|alpha |}leftlangle delta _{a},partial ^{alpha }varphi rightrangle =(-1)^{|alpha |}partial ^{alpha }varphi (x){Big |}_{x=a}{text{ for all }}varphi in S(U).}

That is, the αth derivative of δ_a is the distribution whose value on any test function φ is the αth derivative of φ at a (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 S is any distribution on U supported on the set {a} consisting of a single point, then there is an integer m and coefficients c_α such that^[45]

S=∑|α|≤mcα∂αδa.{displaystyle S=sum _{|alpha |leq m}c_{alpha }partial ^{alpha }delta _{a}.}

Representations of the delta function

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

δ(x)=limε→0+ηε(x),{displaystyle delta (x)=lim _{varepsilon to 0^{+}}eta _{varepsilon }(x),}

where η_ε(x) is sometimes called a nascent delta function. This limit is meant in a weak sense: either that

limε→0+∫−∞∞ηε(x)f(x)dx=f(0){displaystyle lim _{varepsilon to 0^{+}}int _{-infty }^{infty }eta _{varepsilon }(x)f(x),dx=f(0)}

(5)

for all continuous functions f having compact support, or that this limit holds for all smooth functions f 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 η_ε can be constructed in the following manner. Let η be an absolutely integrable function on R of total integral 1, and define

ηε(x)=ε−1η(xε).{displaystyle eta _{varepsilon }(x)=varepsilon ^{-1}eta left({frac {x}{varepsilon }}right).}

In n dimensions, one uses instead the scaling

ηε(x)=ε−nη(xε).{displaystyle eta _{varepsilon }(x)=varepsilon ^{-n}eta left({frac {x}{varepsilon }}right).}

Then a simple change of variables shows that η_ε also has integral 1. One may show that (5) holds for all continuous compactly supported functions f,^[46] and so η_ε converges weakly to δ in the sense of measures.

The η_ε constructed in this way are known as an approximation to the identity.^[47] This terminology is because the space L¹(R) of absolutely integrable functions is closed under the operation of convolution of functions: f ∗ g ∈ L¹(R) whenever f and g are in L¹(R). However, there is no identity in L¹(R) for the convolution product: no element h such that f ∗ h = f for all f. Nevertheless, the sequence η_ε does approximate such an identity in the sense that

f∗ηε→fas ε→0.{displaystyle f*eta _{varepsilon }to fquad {text{as }}varepsilon to 0.}

This limit holds in the sense of mean convergence (convergence in L¹). Further conditions on the η_ε, for instance that it be a mollifier associated to a compactly supported function,^[48] are needed to ensure pointwise convergence almost everywhere.

If the initial η = η₁ is itself smooth and compactly supported then the sequence is called a mollifier. The standard mollifier is obtained by choosing η to be a suitably normalized bump function, for instance

η(x)={e−11−|x|2if |x|<10if |x|≥1.{displaystyle eta (x)={begin{cases}e^{-{frac {1}{1-|x|^{2}}}}&{text{if }}|x|<1\0&{text{if }}|x|geq 1.end{cases}}}

In some situations such as numerical analysis, a piecewise linear approximation to the identity is desirable. This can be obtained by taking η₁ to be a hat function. With this choice of η₁, one has

ηε(x)=ε−1max(1−|xε|,0){displaystyle eta _{varepsilon }(x)=varepsilon ^{-1}max left(1-left|{frac {x}{varepsilon }}right|,0right)}

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 η₁ 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 η₁ to be any probability distribution at all, and letting η_ε(x) = η₁(x/ε)/ε as above will give rise to an approximation to the identity. In general this converges more rapidly to a delta function if, in addition, η has mean 0 and has small higher moments. For instance, if η₁ is the uniform distribution on [−1/2, 1/2], also known as the rectangular function, then:^[49]

ηε(x)=1εrect⁡(xε)={1ε,−ε2<x<ε20,otherwise.{displaystyle 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}}}

Another example is with the Wigner semicircle distribution

ηε(x)={2πε2ε2−x2,−ε<x<ε0,otherwise{displaystyle eta _{varepsilon }(x)={begin{cases}{frac {2}{pi varepsilon ^{2}}}{sqrt {varepsilon ^{2}-x^{2}}},&-varepsilon <x<varepsilon \0,&{text{otherwise}}end{cases}}}

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 η_ε with η_δ must satisfy

ηε∗ηδ=ηε+δ{displaystyle eta _{varepsilon }*eta _{delta }=eta _{varepsilon +delta }}

for all ε, δ > 0. Convolution semigroups in L¹ 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

{∂∂tη(t,x)=Aη(t,x),t>0limt→0+η(t,x)=δ(x){displaystyle {begin{cases}{dfrac {partial }{partial t}}eta (t,x)=Aeta (t,x),quad t>0\[5pt]displaystyle lim _{tto 0^{+}}eta (t,x)=delta (x)end{cases}}}

in which the limit is as usual understood in the weak sense. Setting η_ε(x) = η(ε, x) 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

ηε(x)=12πεe−x22ε{displaystyle eta _{varepsilon }(x)={frac {1}{sqrt {2pi varepsilon }}}mathrm {e} ^{-{frac {x^{2}}{2varepsilon }}}}

represents the temperature in an infinite wire at time t > 0, if a unit of heat energy is stored at the origin of the wire at time t = 0. This semigroup evolves according to the one-dimensional heat equation:

∂u∂t=12∂2u∂x2.{displaystyle {frac {partial u}{partial t}}={frac {1}{2}}{frac {partial ^{2}u}{partial x^{2}}}.}

In probability theory, η_ε(x) is a normal distribution of variance ε and mean 0. It represents the probability density at time t = ε 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 Rⁿ, the heat kernel is

ηε=1(2πε)n/2e−x⋅x2ε,{displaystyle eta _{varepsilon }={frac {1}{(2pi varepsilon )^{n/2}}}mathrm {e} ^{-{frac {xcdot x}{2varepsilon }}},}

and has the same physical interpretation, mutatis mutandis. It also represents a nascent delta function in the sense that η_ε → δ in the distribution sense as ε → 0.

The Poisson kernel

The Poisson kernel

ηε(x)=1πεε2+x2=∫−∞∞e2πiξx−|εξ|dξ{displaystyle eta _{varepsilon }(x)={frac {1}{pi }}{frac {varepsilon }{varepsilon ^{2}+x^{2}}}=int _{-infty }^{infty }mathrm {e} ^{2pi mathrm {i} xi x-|varepsilon xi |};dxi }

is the fundamental solution of the Laplace equation in the upper half-plane.^[50] 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. This semigroup evolves according to the equation

∂u∂t=−(−∂2∂x2)12u(t,x){displaystyle {frac {partial u}{partial t}}=-left(-{frac {partial ^{2}}{partial x^{2}}}right)^{frac {1}{2}}u(t,x)}

where the operator is rigorously defined as the Fourier multiplier

F[(−∂2∂x2)12f](ξ)=|2πξ|Ff(ξ).{displaystyle {mathcal {F}}left[left(-{frac {partial ^{2}}{partial x^{2}}}right)^{frac {1}{2}}fright](xi )=|2pi xi |{mathcal {F}}f(xi ).}

Oscillatory integrals

In areas of physics such as wave propagation and wave mechanics, the equations involved are hyperbolic and so may have more singular solutions. As a result, the 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,^[51] is the rescaled Airy function

ε−1/3Ai⁡(xε−1/3).{displaystyle varepsilon ^{-1/3}operatorname {Ai} left(xvarepsilon ^{-1/3}right).}

Although using the Fourier transform, it is easy to see that this generates a semigroup in some sense—it is not absolutely integrable and so cannot define a semigroup in the above strong sense. Many 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 R¹⁺¹:^[52]

c−2∂2u∂t2−Δu=0u=0,∂u∂t=δfor t=0.{displaystyle {begin{aligned}c^{-2}{frac {partial ^{2}u}{partial t^{2}}}-Delta u&=0\u=0,quad {frac {partial u}{partial t}}=delta &qquad {text{for }}t=0.end{aligned}}}

The solution u 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)

ηε(x)=1πxsin⁡(xε)=12π∫−1ε1εcos⁡(kx)dk{displaystyle eta _{varepsilon }(x)={frac {1}{pi x}}sin left({frac {x}{varepsilon }}right)={frac {1}{2pi }}int _{-{frac {1}{varepsilon }}}^{frac {1}{varepsilon }}cos(kx);dk}

and the Bessel function

ηε(x)=1εJ1ε(x+1ε).{displaystyle eta _{varepsilon }(x)={frac {1}{varepsilon }}J_{frac {1}{varepsilon }}left({frac {x+1}{varepsilon }}right).}

Plane wave decomposition

One approach to the study of a linear partial differential equation

L[u]=f,{displaystyle L[u]=f,}

where L is a differential operator on Rⁿ, is to seek first a fundamental solution, which is a solution of the equation

L[u]=δ.{displaystyle L[u]=delta .}

When L is particularly simple, this problem can often be resolved using the Fourier transform directly (as in the case of the Poisson kernel and heat kernel already mentioned). For more complicated operators, it is sometimes easier first to consider an equation of the form

L[u]=h{displaystyle L[u]=h}

where h is a plane wave function, meaning that it has the form

h=h(x⋅ξ){displaystyle h=h(xcdot xi )}

for some vector ξ. Such an equation can be resolved (if the coefficients of L are analytic functions) by the Cauchy–Kovalevskaya theorem or (if the coefficients of L 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).^[53] Choose k so that n + k is an even integer, and for a real number s, put

g(s)=Re⁡[−sklog⁡(−is)k!(2πi)n]={|s|k4k!(2πi)n−1n odd−|s|klog⁡|s|k!(2πi)nn even.{displaystyle g(s)=operatorname {Re} left[{frac {-s^{k}log(-is)}{k!(2pi i)^{n}}}right]={begin{cases}{frac {|s|^{k}}{4k!(2pi i)^{n-1}}}&n{text{ odd}}\[5pt]-{frac {|s|^{k}log |s|}{k!(2pi i)^{n}}}&n{text{ even.}}end{cases}}}

Then δ is obtained by applying a power of the Laplacian to the integral with respect to the unit sphere measure dω of g(x · ξ) for ξ in the unit sphere Sⁿ⁻¹:

δ(x)=Δx(n+k)/2∫Sn−1g(x⋅ξ)dωξ.{displaystyle delta (x)=Delta _{x}^{(n+k)/2}int _{S^{n-1}}g(xcdot xi ),domega _{xi }.}

The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function φ,

φ(x)=∫Rnφ(y)dyΔxn+k2∫Sn−1g((x−y)⋅ξ)dωξ.{displaystyle varphi (x)=int _{mathbf {R} ^{n}}varphi (y),dy,Delta _{x}^{frac {n+k}{2}}int _{S^{n-1}}g((x-y)cdot xi ),domega _{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 φ(x) from its integrals over hyperplanes. For instance, if n is odd and k = 1, then the integral on the right hand side is

cnΔxn+12∫∫Sn−1φ(y)|(y−x)⋅ξ|dωξdy=cnΔx(n+1)/2∫Sn−1dωξ∫−∞∞|p|Rφ(ξ,p+x⋅ξ)dp{displaystyle {begin{aligned}&c_{n}Delta _{x}^{frac {n+1}{2}}int int _{S^{n-1}}varphi (y)|(y-x)cdot xi |,domega _{xi },dy\[5pt]={}&c_{n}Delta _{x}^{(n+1)/2}int _{S^{n-1}},domega _{xi }int _{-infty }^{infty }|p|Rvarphi (xi ,p+xcdot xi ),dpend{aligned}}}

where Rφ(ξ, p) is the Radon transform of φ:

Rφ(ξ,p)=∫x⋅ξ=pf(x)dn−1x.{displaystyle Rvarphi (xi ,p)=int _{xcdot xi =p}f(x),d^{n-1}x.}

An alternative equivalent expression of the plane wave decomposition, from Gel'fand & Shilov (1966–1968, I, §3.10), is

δ(x)=(n−1)!(2πi)n∫Sn−1(x⋅ξ)−ndωξ{displaystyle delta (x)={frac {(n-1)!}{(2pi i)^{n}}}int _{S^{n-1}}(xcdot xi )^{-n},domega _{xi }}

for n even, and

δ(x)=12(2πi)n−1∫Sn−1δ(n−1)(x⋅ξ)dωξ{displaystyle delta (x)={frac {1}{2(2pi i)^{n-1}}}int _{S^{n-1}}delta ^{(n-1)}(xcdot xi ),domega _{xi }}

for n odd.

Fourier kernels

In the study of Fourier series, a major question consists of determining whether and in what sense the Fourier series associated with a periodic function converges to the function. The nth partial sum of the Fourier series of a function f of period 2π is defined by convolution (on the interval [−π,π]) with the Dirichlet kernel:

DN(x)=∑n=−NNeinx=sin⁡((N+12)x)sin⁡(x/2).{displaystyle D_{N}(x)=sum _{n=-N}^{N}e^{inx}={frac {sin left((N+{tfrac {1}{2}})xright)}{sin(x/2)}}.}

Thus,

sN(f)(x)=DN∗f(x)=∑n=−NNaneinx{displaystyle s_{N}(f)(x)=D_{N}*f(x)=sum _{n=-N}^{N}a_{n}e^{inx}}

where

an=12π∫−ππf(y)e−inydy.{displaystyle a_{n}={frac {1}{2pi }}int _{-pi }^{pi }f(y)e^{-iny},dy.}

A fundamental result of elementary Fourier series states that the Dirichlet kernel tends to the a multiple of the delta function as N → ∞. This is interpreted in the distribution sense, that

sN(f)(0)=∫RDN(x)f(x)dx→2πf(0){displaystyle s_{N}(f)(0)=int _{mathbf {R} }D_{N}(x)f(x),dxto 2pi f(0)}

for every compactly supported smooth function f. Thus, formally one has

δ(x)=12π∑n=−∞∞einx{displaystyle delta (x)={frac {1}{2pi }}sum _{n=-infty }^{infty }e^{inx}}

on the interval [−π,π].

In spite of this, the result does not hold for all compactly supported continuous functions: that is D_N 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 in order to produce convergence. The method of Cesàro summation leads to the Fejér kernel^[54]

FN(x)=1N∑n=0N−1Dn(x)=1N(sin⁡Nx2sin⁡x2)2.{displaystyle F_{N}(x)={frac {1}{N}}sum _{n=0}^{N-1}D_{n}(x)={frac {1}{N}}left({frac {sin {frac {Nx}{2}}}{sin {frac {x}{2}}}}right)^{2}.}

The Fejér kernels tend to the delta function in a stronger sense that^[55]

∫RFN(x)f(x)dx→2πf(0){displaystyle int _{mathbf {R} }F_{N}(x)f(x),dxto 2pi f(0)}

for every compactly supported continuous function f. 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 support functions are dense in L², and the action of the delta distribution on such functions is well-defined. In many applications, it is possible to identify subspaces of L² 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 R implies that any square-integrable function f such that

‖f‖H12=∫−∞∞|f^(ξ)|2(1+|ξ|2)dξ<∞{displaystyle |f|_{H^{1}}^{2}=int _{-infty }^{infty }|{widehat {f}}(xi )|^{2}(1+|xi |^{2}),dxi <infty }

is automatically continuous, and satisfies in particular

δ[f]=|f(0)|<C‖f‖H1.{displaystyle delta [f]=|f(0)|<C|f|_{H^{1}}.}

Thus δ is a bounded linear functional on the Sobolev space H¹. Equivalently δ is an element of the continuous dual space H⁻¹ of H¹. More generally, in n dimensions, one has δ ∈ H^−s(Rⁿ) provided s > n / 2.

Spaces of holomorphic functions

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

f(z)=12πi∮∂D⁡f(ζ)dζζ−z,z∈D{displaystyle f(z)={frac {1}{2pi i}}oint _{partial D}{frac {f(zeta ),dzeta }{zeta -z}},quad zin D}

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

δz[f]=f(z)=12πi∮∂D⁡f(ζ)dζζ−z.{displaystyle delta _{z}[f]=f(z)={frac {1}{2pi i}}oint _{partial D}{frac {f(zeta ),dzeta }{zeta -z}}.}

Moreover, let H²(∂D) be the Hardy space consisting of the closure in L²(∂D) of all holomorphic functions in D continuous up to the boundary of D. Then functions in H²(∂D) uniquely extend to holomorphic functions in D, and the Cauchy integral formula continues to hold. In particular for z ∈ D, the delta function δ_z is a continuous linear functional on H²(∂D). This is a special case of the situation in several complex variables in which, for smooth domains D, the Szegő kernel plays the role of the Cauchy integral.

Resolutions of the identity

Given a complete orthonormal basis set of functions {φ_n} in a separable Hilbert space, for example, the normalized eigenvectors of a compact self-adjoint operator, any vector f can be expressed as:

f=∑n=1∞αnφn.{displaystyle f=sum _{n=1}^{infty }alpha _{n}varphi _{n}.}

The coefficients {α_n} are found as:

αn=⟨φn,f⟩,{displaystyle alpha _{n}=langle varphi _{n},frangle ,}

which may be represented by the notation:

αn=φn†f,{displaystyle alpha _{n}=varphi _{n}^{dagger }f,}

a form of the bra–ket notation of Dirac.^[56] Adopting this notation, the expansion of f takes the dyadic form:^[57]

f=∑n=1∞φn(φn†f).{displaystyle f=sum _{n=1}^{infty }varphi _{n}left(varphi _{n}^{dagger }fright).}

Letting I denote the identity operator on the Hilbert space, the expression

I=∑n=1∞φnφn†,{displaystyle I=sum _{n=1}^{infty }varphi _{n}varphi _{n}^{dagger },}

is called a resolution of the identity. When the Hilbert space is the space L²(D) of square-integrable functions on a domain D, the quantity:

φnφn†,{displaystyle varphi _{n}varphi _{n}^{dagger },}

is an integral operator, and the expression for f can be rewritten as:

f(x)=∑n=1∞∫D(φn(x)φn∗(ξ))f(ξ)dξ.{displaystyle f(x)=sum _{n=1}^{infty }int _{D},left(varphi _{n}(x)varphi _{n}^{*}(xi )right)f(xi ),dxi .}

The right-hand side converges to f in the L² sense. It need not hold in a pointwise sense, even when f is a continuous function. Nevertheless, it is common to abuse notation and write

f(x)=∫δ(x−ξ)f(ξ)dξ,{displaystyle f(x)=int ,delta (x-xi )f(xi ),dxi ,}

resulting in the representation of the delta function:^[58]

δ(x−ξ)=∑n=1∞φn(x)φn∗(ξ).{displaystyle delta (x-xi )=sum _{n=1}^{infty }varphi _{n}(x)varphi _{n}^{*}(xi ).}

With a suitable rigged Hilbert space (Φ, L²(D), Φ*) where Φ ⊂ L²(D) contains all compactly supported smooth functions, this summation may converge in Φ*, depending on the properties of the basis φ_n. 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.^[59]

Infinitesimal delta functions

Cauchy used an infinitesimal α to write down a unit impulse, infinitely tall and narrow Dirac-type delta function δ_α satisfying ∫F(x)δα(x)=F(0){displaystyle int F(x)delta _{alpha }(x)=F(0)} in a number of articles in 1827.^[60] 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 Yamashita (2007) 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 F one has ∫F(x)δα(x)=F(0){displaystyle int F(x)delta _{alpha }(x)=F(0)} as anticipated by Fourier and Cauchy.

Dirac comb

A Dirac comb is an infinite series of Dirac delta functions spaced at intervals of T

A so-called uniform "pulse train" of Dirac delta measures, which is known as a Dirac comb, or as the Shah 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,

Δ(x)=∑n=−∞∞δ(x−n),{displaystyle Delta (x)=sum _{n=-infty }^{infty }delta (x-n),}

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 f is any Schwartz function, then the periodization of f is given by the convolution

(f∗Δ)(x)=∑n=−∞∞f(x−n).{displaystyle (f*Delta )(x)=sum _{n=-infty }^{infty }f(x-n).}

In particular,

(f∗Δ)∧=f^Δ^=f^Δ{displaystyle (f*Delta )^{wedge }={widehat {f}}{widehat {Delta }}={widehat {f}}Delta }

is precisely the Poisson summation formula.^[61]

Sokhotski–Plemelj theorem

The Sokhotski–Plemelj theorem, important in quantum mechanics, relates the delta function to the distribution p.v. 1/x, the Cauchy principal value of the function 1/x, defined by

⟨p.v.⁡1x,φ⟩=limε→0+∫|x|>εφ(x)xdx.{displaystyle leftlangle operatorname {p.v.} {frac {1}{x}},varphi rightrangle =lim _{varepsilon to 0^{+}}int _{|x|>varepsilon }{frac {varphi (x)}{x}},dx.}

Sokhotsky's formula states that^[62]

limε→0+1x±iε=p.v.⁡1x∓iπδ(x),{displaystyle lim _{varepsilon to 0^{+}}{frac {1}{xpm ivarepsilon }}=operatorname {p.v.} {frac {1}{x}}mp ipi delta (x),}

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

limε→0+∫−∞∞f(x)x±iεdx=∓iπf(0)+limε→0+∫|x|>εf(x)xdx.{displaystyle lim _{varepsilon to 0^{+}}int _{-infty }^{infty }{frac {f(x)}{xpm ivarepsilon }},dx=mp ipi f(0)+lim _{varepsilon to 0^{+}}int _{|x|>varepsilon }{frac {f(x)}{x}},dx.}

Relationship to the Kronecker delta

The Kronecker delta δ_ij is the quantity defined by

δij={1i=j0i≠j{displaystyle delta _{ij}={begin{cases}1&i=j\0&inot =jend{cases}}}

for all integers i, j. This function then satisfies the following analog of the sifting property: if (ai)i∈Z{displaystyle (a_{i})_{iin mathbf {Z} }} is any doubly infinite sequence, then

∑i=−∞∞aiδik=ak.{displaystyle sum _{i=-infty }^{infty }a_{i}delta _{ik}=a_{k}.}

Similarly, for any real or complex valued continuous function f on R, the Dirac delta satisfies the sifting property

∫−∞∞f(x)δ(x−x0)dx=f(x0).{displaystyle int _{-infty }^{infty }f(x)delta (x-x_{0}),dx=f(x_{0}).}

This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function.^[63]

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 fully continuous distributions). For example, the probability density function f(x) of a discrete distribution consisting of points x = {x₁, ..., x_n}, with corresponding probabilities p₁, ..., p_n, can be written as

f(x)=∑i=1npiδ(x−xi).{displaystyle f(x)=sum _{i=1}^{n}p_{i}delta (x-x_{i}).}

As another example, consider a distribution 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

f(x)=0.612πe−x22+0.4δ(x−3.5).{displaystyle f(x)=0.6,{frac {1}{sqrt {2pi }}}e^{-{frac {x^{2}}{2}}}+0.4,delta (x-3.5).}

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 B(t) is given by

ℓ(x,t)=∫0tδ(x−B(s))ds{displaystyle ell (x,t)=int _{0}^{t}delta (x-B(s)),ds}

and represents the amount of time that the process spends at the point x in the range of the process. More precisely, in one dimension this integral can be written

ℓ(x,t)=limε→0+12ε∫0t1[x−ε,x+ε](B(s))ds{displaystyle ell (x,t)=lim _{varepsilon to 0^{+}}{frac {1}{2varepsilon }}int _{0}^{t}mathbf {1} _{[x-varepsilon ,x+varepsilon ]}(B(s)),ds}

where 1_{[x−ε, x+ε]} is the indicator function of the interval [x−ε, x+ε].

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 L² 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 {φ_n} of wave functions is orthonormal if they are normalized by

⟨φn∣φm⟩=δnm{displaystyle langle varphi _{n}mid varphi _{m}rangle =delta _{nm}}

where δ here refers to the Kronecker delta. A set of orthonormal wave functions is complete in the space of square-integrable functions if any wave function ψ can be expressed as a combination of the φ_n:

ψ=∑cnφn,{displaystyle psi =sum c_{n}varphi _{n},}

with cn=⟨φn|ψ⟩{displaystyle c_{n}=langle varphi _{n}|psi rangle }. 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:

I=∑|φn⟩⟨φn|.{displaystyle I=sum |varphi _{n}rangle langle varphi _{n}|.}

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, Qψ(x) = xψ(x). 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 by an appropriate rigged Hilbert space.^[64] In this context, the position operator has a complete set of eigen-distributions, labeled by the points y of the real line, given by

φy(x)=δ(x−y).{displaystyle varphi _{y}(x)=delta (x-y).}

The eigenfunctions of position are denoted by φy=|y⟩{displaystyle varphi _{y}=|yrangle } 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 P on the Hilbert space, provided the spectrum of P is continuous and there are no degenerate eigenvalues. In that case, there is a set Ω of real numbers (the spectrum), and a collection φ_y of distributions indexed by the elements of Ω, such that

Pφy=yφy.{displaystyle Pvarphi _{y}=yvarphi _{y}.}

That is, φ_y are the eigenvectors of P. If the eigenvectors are normalized so that

⟨φy,φy′⟩=δ(y−y′){displaystyle langle varphi _{y},varphi _{y'}rangle =delta (y-y')}

in the distribution sense, then for any test function ψ,

ψ(x)=∫Ωc(y)φy(x)dy{displaystyle psi (x)=int _{Omega }c(y)varphi _{y}(x),dy}

where

c(y)=⟨ψ,φy⟩.{displaystyle c(y)=langle psi ,varphi _{y}rangle .}

That is, as in the discrete case, there is a resolution of the identity

I=∫Ω|φy⟩⟨φy|dy{displaystyle I=int _{Omega }|varphi _{y}rangle ,langle varphi _{y}|,dy}

where the operator-valued integral is again understood in the weak sense. If the spectrum of P has both continuous and discrete parts, then the resolution of the identity involves a summation over the discrete spectrum and 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 I at time t = 0 can be written

md2ξdt2+kξ=Iδ(t),{displaystyle m{frac {mathrm {d} ^{2}xi }{mathrm {d} t^{2}}}+kxi =Idelta (t),}

where m is the mass, ξ the deflection and k the spring constant.

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

EId4wdx4=q(x),{displaystyle EI{frac {mathrm {d} ^{4}w}{mathrm {d} x^{4}}}=q(x),}

where EI is the bending stiffness of the beam, w the deflection, x the spatial coordinate and q(x) the load distribution. If a beam is loaded by a point force F at x = x₀, the load distribution is written

q(x)=Fδ(x−x0).{displaystyle q(x)=Fdelta (x-x_{0}).}

As 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 F at a distance d apart. They then produce a moment M = Fd acting on the beam. Now, let the distance d approach the limit zero, while M is kept constant. The load distribution, assuming a clockwise moment acting at x = 0, is written

q(x)=limd→0(Fδ(x)−Fδ(x−d))=limd→0(Mdδ(x)−Mdδ(x−d))=Mlimd→0δ(x)−δ(x−d)d=Mδ′(x).{displaystyle {begin{aligned}q(x)&=lim _{dto 0}{Big (}Fdelta (x)-Fdelta (x-d){Big )}\[4pt]&=lim _{dto 0}left({frac {M}{d}}delta (x)-{frac {M}{d}}delta (x-d)right)\[4pt]&=Mlim _{dto 0}{frac {delta (x)-delta (x-d)}{d}}\[4pt]&=Mdelta '(x).end{aligned}}}

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)


• Delta potential


• Dirac measure


• Fundamental solution


• Green's function


• Laplacian of the indicator

Notes

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External links

• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Delta-function", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• 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.