Weierstrass transform on arvau

Wikipedia / Weierstrass transform

In mathematics, the Weierstrass transform of a function, named after Karl Weierstrass.

defined by,

which is the convolution with the Gaussian function

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.

, when the defining integral fails to converge.

The Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation describes the initial temperature at each point of an infinitely long rod that has constant thermal conductivity different from 1, we can define the generalized Weierstrass transform.

arbitrarily well with analytic functions.

Names

Weierstrass used this transform in his original proof of the Weierstrass approximation theorem. It is also known as the Gauss transform or Gauss–Weierstrass transform after Carl Friedrich Gauss and as the Hille transform after Einar Carl Hille mentioned below is known in signal analysis as a Gaussian filter and in image processing) as a Gaussian blur.

Transforms of some important functions

Constant Functions

Every constant function is its own Weierstrass transform.

Polynomials

The Weierstrass transform of any polynomial is a polynomial of the same degree, and in fact has the same leading coefficient (the asymptotic growth denotes the (physicist's) Hermite polynomial. 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.

Exponentials, Sines, and Cosines

is thus an eigenfunction.

is the imaginary unit, and applying Euler's identity.

Gaussian Functions

is a Gaussian function and its Weierstrass transform is also a Gaussian function, but a "wider" one.

General properties

; this assignment is linear. 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 will exist for all complex and forms a holomorphic function on that strip of the complex plane mentioned above.

are equal. 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.

-th derivative.

There is a formula relating the Weierstrass transform W and the two-sided Laplace transform. If we define

then

Low-pass filter

. In terms of signal analysis will contain the same frequency, but with an amplitude. This has the consequence that higher frequencies are reduced more than lower ones, and the Weierstrass transform thus acts as a low-pass filter. This can also be shown with the continuous Fourier transform, as follows. The Fourier transform analyzes a signal in terms of its frequencies, transforms convolutions into products, and transforms Gaussians into Gaussians. The Weierstrass transform is convolution with a Gaussian and is therefore multiplication of the Fourier transformed signal with a Gaussian, followed by application of the inverse Fourier transform. This multiplication with a Gaussian in frequency space blends out high frequencies, which is another way of describing the "smoothing" property of the Weierstrass transform.

The inverse transform

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

, where the operator on the right is to be understood as acting on the function f(x) as

cannot be meaningfully defined.

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.

One may, alternatively, attempt to invert the Weierstrass transform in a slightly different way: given the analytic function

to obtain

once more using a fundamental property of the (physicists') Hermite polynomials.

as a series of Hermite polynomials.

A third method of inverting the Weierstrass transform exploits its connection to the Laplace transform mentioned above, and the well-known inversion formula for the Laplace transform. The result is stated below for distributions.

Generalizations

thus defining an operator , the generalized Weierstrass transform.

to be the identity operator (i.e. convolution with the Dirac delta function), and these then form a one-parameter semigroup of operators.

used for the generalized Weierstrass transform is sometimes called the Gauss–Weierstrass kernel, and is Green's function for the diffusion equation a consequence of the substitution rule.

The Weierstrass transform can also be defined for certain classes of distributionss or "generalized functions". For example, the Weierstrass transform of the Dirac delta is any fixed real number, and the limit is to be taken in the sense of distributions.

as the square of the Euclidean length; the factor in front of the integral has to be adjusted so that the Gaussian will have a total integral of 1.

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.

Related transforms

instead of with a Gaussian, one obtains the Poisson transform which smoothes and averages a given function in a manner similar to the Weierstrass transform.