Dirac delta function
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In
The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1. The mathematical rigor of the delta function was disputed until Laurent Schwartz developed the theory of distributions, where it is defined as a linear form acting on functions.
Motivation and overview
The
To be specific, suppose that a billiard ball is at rest. At time it is struck by another ball, imparting it with a momentum P, 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 P δ(t); the units of δ(t) 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 t is found by integration:
Now, the model situation of an instantaneous transfer of momentum requires taking the limit as Δt → 0, giving a result everywhere except at 0:
Here the functions 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) 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
which holds for all , should continue to hold in the limit. So, in the equation , 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
The Dirac delta is not truly a function, at least not a usual one with domain and range in
History
which is tantamount to the introduction of the δ-function in the form:[7]
Later,
Cauchy pointed out that in some circumstances the order of integration is significant in this result (contrast Fubini's theorem).[10][11]
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
where the δ-function is expressed as
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:[12]
- 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 rapidlyto zero (in the neighborhood of infinity) to ensure the existence of the Fourier integral. For example, the Fourier transform of such simple functions as polynomials does not exist in the classical sense. The extension of the classical Fourier transformation to distributions considerably enlarged the class of functions that could be transformed and this removed many obstacles.
Further developments included generalization of the Fourier integral, "beginning with Plancherel's pathbreaking L2-theory (1910), continuing with Wiener's and Bochner's works (around 1930) and culminating with the amalgamation into L. Schwartz's theory of distributions (1945) ...",[13] and leading to the formal development of the Dirac delta function.
An
Definitions
The Dirac delta function can be loosely thought of as a function on the real line which is zero everywhere except at the origin, where it is infinite,
and which is also constrained to satisfy the identity[17]
This is merely a
As a measure
One way to rigorously capture the notion of the Dirac delta function is to define a
for all continuous compactly supported functions f. The measure δ is not
holds.
As a
This means that H(x) is the integral of the cumulative indicator function 1(−∞, x] with respect to the measure δ; to wit,
the latter being the measure of this interval; more formally, δ((−∞, x]). Thus in particular the integration of the delta function against a continuous function can be properly understood as a Riemann–Stieltjes integral:[22]
All higher
As a distribution
In the theory of distributions, a generalized function is considered not a function in itself but only through how it affects other functions when "integrated" against them.[23] 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
-
(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 MN and a constant CN such that for every test function φ, one has the inequality[25]
where sup represents the supremum. With the δ distribution, one has such an inequality (with CN = 1) with MN = 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 several equivalent ways. For instance, it is the
Intuitively, if integration by parts were permitted, then the latter integral should simplify to
and indeed, a form of integration by parts is permitted for the Stieltjes integral, and in that case, one does have
In the context of measure theory, the Dirac measure gives rise to 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
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, ..., xn), one has[26]
-
(2)
The delta function can also be defined in the sense of distributions exactly as above in the one-dimensional case.[27] 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.[28][29]
The notion of a
is the delta measure or unit mass concentrated at x0.
Another common generalization of the delta function is to a
-
(3)
for all compactly supported smooth real-valued functions φ on M.[31] A common special case of this construction is a case in which M is an open set in the Euclidean space Rn.
On a
Properties
Scaling and symmetry
The delta function satisfies the following scaling property for a non-zero scalar α:[34]
and so
-
(4)
Scaling property proof:
In particular, the delta function is an
which is homogeneous of degree −1.
Algebraic properties
The distributional product of δ with x is equal to zero:
More generally, for all positive integers .
Conversely, if xf(x) = xg(x), where f and g are distributions, then
for some constant c.[35]
Translation
The integral of the time-delayed Dirac delta is[36]
This is sometimes referred to as the sifting property[37] or the sampling property.[38] The delta function is said to "sift out" the value of f(t) at t = T.[39]
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:
The sifting property 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)
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
provided that g is a
It is natural therefore to define the composition δ(g(x)) for continuously differentiable functions g by
where the sum extends over all roots (i.e., all the different ones) of g(x), which are assumed to be
In the integral form, the generalized scaling property may be written as
Indefinite integral
For a constant and a "well-behaved" arbitrary real-valued function y(x),
Properties in n dimensions
The delta distribution in an n-dimensional space satisfies the following scaling property instead,
Under any reflection or rotation ρ, the delta function is invariant,
As in the one-variable case, it is possible to define the composition of δ with a
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[42]
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:
where σ is the hypersurface measure associated to S. This generalization is associated with the
where n is the outward normal.[43][44] For a proof, see e.g. the article on the surface delta function.
In three dimensions, the delta function is represented in spherical coordinates by:
Fourier transform
The delta function is a tempered distribution, and therefore it has a well-defined Fourier transform. Formally, one finds[45]
Properly speaking, the Fourier transform of a distribution is defined by imposing self-adjointness of the Fourier transform under the duality pairing of tempered distributions with
for all Schwartz functions φ. And indeed it follows from this that
As a result of this identity, the convolution of the delta function with any other tempered distribution S is simply S:
That is to say that δ is an
The inverse Fourier transform of the tempered distribution f(ξ) = 1 is the delta function. Formally, this is expressed as
In these terms, the delta function provides a suggestive statement of the orthogonality property of the Fourier kernel on R. Formally, one has
This is, of course, shorthand for the assertion that the Fourier transform of the tempered distribution
By analytic continuation of the Fourier transform, the Laplace transform of the delta function is found to be[46]
Derivatives
The derivative of the Dirac delta distribution, denoted δ′ and also called the Dirac delta prime or Dirac delta derivative as described in Laplacian of the indicator, is defined on compactly supported smooth test functions φ by[47]
The first equality here is a kind of integration by parts, for if δ were a true function then
By mathematical induction, the k-th derivative of δ is defined similarly as the distribution given on test functions by
In particular, δ is an infinitely differentiable distribution.
The first derivative of the delta function is the distributional limit of the difference quotients:[48]
More properly, one has
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.[49]
The derivative of the delta function satisfies a number of basic properties, including:[50]
The latter of these properties can also be demonstrated by applying distributional derivative definition, Liebniz 's theorem and linearity of inner product:[51]
Furthermore, the convolution of δ′ with a compactly-supported, smooth function f is
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 , the Dirac delta distribution centered at a point a ∈ U is defined by[52]
That is, the α-th derivative of δa is the distribution whose value on any test function φ is the α-th derivative of φ at a (with the appropriate positive or negative sign).
The first partial derivatives of the delta function are thought of as
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 cα such that[52][53]
Representations of the delta function
The delta function can be viewed as the limit of a sequence of functions
where ηε(x) is sometimes called a nascent delta function. This limit is meant in a weak sense: either that
-
(5)
for all
Approximations to the identity
Typically a nascent delta function ηε can be constructed in the following manner. Let η be an absolutely integrable function on R of total integral 1, and define
In n dimensions, one uses instead the scaling
Then a simple change of variables shows that ηε also has integral 1. One may show that (5) holds for all continuous compactly supported functions f,[54] and so ηε converges weakly to δ in the sense of measures.
The ηε constructed in this way are known as an approximation to the identity.[55] 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 ηε does approximate such an identity in the sense that
This limit holds in the sense of
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
In some situations such as
which are all continuous and compactly supported, although not smooth and so not a mollifier.
Probabilistic considerations
In the context of
Another example is with the Wigner semicircle distribution
This is continuous and compactly supported, but not a mollifier because it is not smooth.
Semigroups
Nascent delta functions often arise as convolution semigroups.[58] This amounts to the further constraint that the convolution of ηε with ηδ must satisfy
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
in which the limit is as usual understood in the weak sense. Setting ηε(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
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:
In probability theory, ηε(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
The Poisson kernel
The Poisson kernel
is the fundamental solution of the
where the operator is rigorously defined as the
Oscillatory integrals
In areas of physics such as
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:[62]
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)
and the Bessel function
Plane wave decomposition
One approach to the study of a linear partial differential equation
where L is a differential operator on Rn, is to seek first a fundamental solution, which is a solution of the equation
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
where h is a plane wave function, meaning that it has the form
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).[63] Choose k so that n + k is an even integer, and for a real number s, put
Then δ is obtained by applying a power of the
Sn−1:The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function φ,
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
where Rφ(ξ, p) is the Radon transform of φ:
An alternative equivalent expression of the plane wave decomposition is:[64]
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 n-th 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:
Despite this, the result does not hold for all compactly supported continuous functions: that is DN 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
The Fejér kernels tend to the delta function in a stronger sense that[66]
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
Sobolev spaces
The
is automatically continuous, and satisfies in particular
Thus δ is a bounded linear functional on the Sobolev space H1. Equivalently δ is an element of the
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
for all holomorphic functions f in D that are continuous on the closure of D. As a result, the delta function δz is represented in this class of holomorphic functions by the Cauchy integral:
Moreover, let H2(∂D) be the
Resolutions of the identity
Given a complete
Letting I denote the
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:
is an integral operator, and the expression for f can be rewritten
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
resulting in the representation of the delta function:[70]
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 φn. 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.[71]
Infinitesimal delta functions
Dirac comb
A so-called uniform "pulse train" of Dirac delta measures, which is known as a
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
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
Sokhotsky's formula states that[75]
Here the limit is understood in the distribution sense, that for all compactly supported smooth functions f,
Relationship to the Kronecker delta
The Kronecker delta δij is the quantity defined by
for all integers i, j. This function then satisfies the following analog of the sifting property: if ai (for i in the set of all integers) is any
Similarly, for any real or complex valued continuous function f on R, the Dirac delta satisfies the sifting property
This exhibits the Kronecker delta function as a discrete analog of the Dirac delta function.[76]
Applications
Probability theory
In
As another example, consider a distribution in which 6/10 of the time returns a standard normal distribution, and 4/10 of the time returns exactly the value 3.5 (i.e. a partly continuous, partly discrete mixture distribution). The density function of this distribution can be written as
The delta function is also used to represent the resulting probability density function of a random variable that is transformed by continuously differentiable function. If Y = g(X) is a continuous differentiable function, then the density of Y can be written as
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
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 {|φn⟩} of wave functions is orthonormal if they are normalized by
where δ is 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 linear combination of the {|φn⟩} with complex coefficients:
with cn = ⟨φn|ψ⟩. 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:
Here the eigenvalues are assumed to be discrete, but the set of eigenvalues of an
The eigenfunctions of position are denoted by φy = |y⟩ 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 φy of distributions indexed by the elements of Ω, such that
That is, φy are the eigenvectors of P. If the eigenvectors are normalized so that
in the distribution sense, then for any test function ψ,
where c(y) = ⟨ψ, φy⟩. That is, as in the discrete case, there is a resolution of the identity
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
where m is the mass, ξ is the deflection, and k is the
As another example, the equation governing the static deflection of a slender
where EI is the bending stiffness of the beam, w is the deflection, x is the spatial coordinate, and q(x) is the load distribution. If a beam is loaded by a point force F at x = x0, the load distribution is written
As the integration of the delta function results in the Heaviside step function, it follows that the static deflection of a slender beam subject to multiple point loads is described by a set of piecewise polynomials.
Also, a point moment acting on a beam can be described by delta functions. Consider two opposing point forces 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
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
Notes
- ^ atis 2013, unit impulse.
- ^ Arfken & Weber 2000, p. 84.
- ^ a b Dirac 1930, §22 The δ function.
- ^ Gelfand & Shilov 1966–1968, Volume I, §1.1.
- ISBN 978-0-08-054996-5.
- ^ Fourier, JB (1822). The Analytical Theory of Heat (English translation by Alexander Freeman, 1878 ed.). The University Press. p. [1]., cf. https://books.google.com/books?id=-N8EAAAAYAAJ&pg=PA449 and pp. 546–551. Original French text.
- ISBN 978-981-238-161-3.
- ISBN 978-0-8176-4393-5.
- ISBN 978-1-58488-575-7.
- ISBN 978-3-7643-2238-0.
- ^
See, for example, Cauchy, Augustin-Louis (1789-1857) Auteur du texte (1882–1974). "Des intégrales doubles qui se présentent sous une forme indéterminèe". Oeuvres complètes d'Augustin Cauchy. Série 1, tome 1 / publiées sous la direction scientifique de l'Académie des sciences et sous les auspices de M. le ministre de l'Instruction publique...
{{cite book}}
: CS1 maint: numeric names: authors list (link) - ISBN 978-0-582-24694-2.
- ISBN 978-9971-5-0666-7.
- ^ Laugwitz 1989, p. 230.
- ^ A more complete historical account can be found in van der Pol & Bremmer 1987, §V.4.
- S2CID 122855515.
- ^ Gelfand & Shilov 1966–1968, Volume I, §1.1, p. 1.
- ^ Dirac 1930, p. 63.
- ^ Rudin 1966, §1.20
- ^ Hewitt & Stromberg 1963, §19.61.
- ^ Driggers 2003, p. 2321 See also Bracewell 1986, Chapter 5 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.
- ^ Hewitt & Stromberg 1963, §9.19.
- ^ Hazewinkel 2011, p. 41.
- ^ Strichartz 1994, §2.2.
- ^ Hörmander 1983, Theorem 2.1.5.
- ^ Bracewell 1986, Chapter 5.
- ^ Hörmander 1983, §3.1.
- ^ Strichartz 1994, §2.3.
- ^ Hörmander 1983, §8.2.
- ^ Rudin 1966, §1.20.
- ^ Dieudonné 1972, §17.3.3.
- ISBN 978-0-8176-4679-0.
- ^ Federer 1969, §2.5.19.
- ^ Strichartz 1994, Problem 2.6.2.
- ^ Vladimirov 1971, Chapter 2, Example 3(d).
- ISBN 978-1-4665-6583-8.
- ^ Weisstein, Eric W. "Sifting Property". MathWorld.
- ISBN 978-0-9709511-6-8.
- ISBN 978-1-4831-4556-3.
- ^ Gelfand & Shilov 1966–1968, Vol. 1, §II.2.5.
- ^ Further refinement is possible, namely to submersions, although these require a more involved change of variables formula.
- ^ Hörmander 1983, §6.1.
- ^ Lange 2012, pp.29–30.
- ^ Gelfand & Shilov 1966–1968, p. 212.
- ^ The numerical factors depend on the conventions for the Fourier transform.
- ^ Bracewell 1986.
- ^ Gelfand & Shilov 1966–1968, p. 26.
- ^ Gelfand & Shilov 1966–1968, §2.1.
- ^ Weisstein, Eric W. "Doublet Function". MathWorld.
- ^ Bracewell 2000, p. 86.
- ^ "Gugo82's comment on the distributional derivative of Dirac's delta". matematicamente.it. 12 September 2010.
- ^ a b c Hörmander 1983, p. 56.
- ^ Rudin 1991, Theorem 6.25.
- ^ Stein & Weiss 1971, Theorem 1.18.
- ^ Rudin 1991, §II.6.31.
- ^ More generally, one only needs η = η1 to have an integrable radially symmetric decreasing rearrangement.
- ^ Saichev & Woyczyński 1997, §1.1 The "delta function" as viewed by a physicist and an engineer, p. 3.
- ISBN 978-1-4939-0258-3.
- ^ Stein & Weiss 1971, §I.1.
- ISBN 978-1-86239-208-3.
- ^ Vallée & Soares 2004, §7.2.
- ^ Hörmander 1983, §7.8.
- ^ Courant & Hilbert 1962, §14.
- ^ Gelfand & Shilov 1966–1968, I, §3.10.
- ^ Lang 1997, p. 312.
- ^ In the terminology of Lang (1997), the Fejér kernel is a Dirac sequence, whereas the Dirichlet kernel is not.
- ^ Hazewinkel 1995, p. 357.
- ^ The development of this section in bra–ket notation is found in (Levin 2002, Coordinate-space wave functions and completeness, pp.=109ff)
- ^ Davis & Thomson 2000, Perfect operators, p.344.
- ^ Davis & Thomson 2000, Equation 8.9.11, p. 344.
- ^ de la Madrid, Bohm & Gadella 2002.
- ^ Laugwitz 1989.
- ^ Córdoba 1988.
- ^ Hörmander 1983, §7.2.
- ^ Vladimirov 1971, §5.7.
- ^ Hartmann 1997, pp. 154–155.
- ^ Isham 1995, §6.2.
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External links
- Media related to Dirac distribution at Wikimedia Commons
- "Delta-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- 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. Archived 2008-03-07 at the Wayback Machine