Dirac delta function
The Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1, is the discrete analog of the Dirac delta function.
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,
The Dirac delta function is the hyperreal function on the reals given by
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 particular, the delta function is an even distribution, in the sense that
In the integral form, the generalized scaling property may be written as
The delta distribution in an n-dimensional space satisfies the following scaling property instead,
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
which again follows by imposing self-adjointness of the Fourier transform.
which can be shown by applying a test function and integrating by parts.
which follows from the properties of the distributional derivative of a convolution.
The delta function can be viewed as the limit of a sequence of functions
which are all continuous and compactly supported, although not smooth and so not a mollifier.
This is continuous and compactly supported, but not a mollifier because it is not smooth.
Some examples of physically important convolution semigroups arising from such a fundamental solution include the following.
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.
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)
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
The Laplacian here is interpreted as a weak derivative, so that this equation is taken to mean that, for any test function φ,
An alternative equivalent expression of the plane wave decomposition, from Gelfand & Shilov (1966–1968, I, §3.10), is
Here the limit is understood in the distribution sense, that for all compactly supported smooth functions f,
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
That is, as in the discrete case, there is a resolution of the identity
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.