Weierstrass transform
is chosen so that the Gaussian will have a total integral of 1, with the consequence that constant functions are not changed by the Weierstrass transform.
The Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient).
describes the initial temperature at each point of an infinitely long rod that has constant thermal conductivity equal to 1, then the temperature distribution of the rod
The generalized Weierstrass transform provides a means to approximate a given integrable function
mentioned below is known in signal analysis as a Gaussian filter and in image processing (when implemented on
denotes the (physicist's) Hermite polynomial of degree
This can be shown by exploiting the fact that the generating function for the Hermite polynomials is closely related to the Gaussian kernel used in the definition of the Weierstrass transform.
is the imaginary unit, and applying Euler's identity, one sees that the Weierstrass transform of the function
Both of these facts are more generally true for any integral transform defined via convolution.
, then it also exists for all real values in between and forms an analytic function there; moreover,
and forms a holomorphic function on that strip of the complex plane.
This expresses the physical fact that the total thermal energy or heat is conserved by the heat equation, or that the total amount of diffusing material is conserved by the diffusion equation.
The Weierstrass transform consequently yields a bounded operator
This has the consequence that higher frequencies are reduced more than lower ones, and the Weierstrass transform thus acts as a low-pass filter.
The following formula, closely related to the Laplace transform of a Gaussian function, and a real analogue to the Hubbard–Stratonovich transformation, is relatively easy to establish: Now replace u with the formal differentiation operator D = d/dx and utilize the Lagrange shift operator (a consequence of the Taylor series formula and the definition of the exponential function), to obtain to thus obtain the following formal expression for the Weierstrass transform
where the operator on the right is to be understood as acting on the function f(x) as The above formal derivation glosses over details of convergence, and the formula
Nevertheless, the rule is still quite useful and can, for example, be used to derive the Weierstrass transforms of polynomials, exponential and trigonometric functions mentioned above.
The formal inverse of the Weierstrass transform is thus given by Again, this formula is not universally valid but can serve as a guide.
It can be shown to be correct for certain classes of functions if the right-hand side operator is properly defined.
[2] One may, alternatively, attempt to invert the Weierstrass transform in a slightly different way: given the analytic function apply
to obtain once more using a fundamental property of the (physicists') Hermite polynomials
is at best formal, since one didn't check whether the final series converges.
thus defining an operator Wt, the generalized Weierstrass transform.
used for the generalized Weierstrass transform is sometimes called the Gauss–Weierstrass kernel, and is Green's function for the diffusion equation
The Weierstrass transform can also be defined for certain classes of distributions or "generalized functions".
In this context, rigorous inversion formulas can be proved, e.g.,
exists, the integral extends over the vertical line in the complex plane with real part
We use the same convolution formula as above but interpret the integral as extending over all of
More generally, the Weierstrass transform can be defined on any Riemannian manifold: the heat equation can be formulated there (using the manifold's Laplace–Beltrami operator), and the Weierstrass transform
is then given by following the solution of the heat equation for one time unit, starting with the initial "temperature distribution"
