Weierstrass transform

In mathematics, the Weierstrass transform of a function f : R → R, named after Karl Weierstrass, is a "smoothed" version of f(x) obtained by averaging the values of f, weighted with a Gaussian centered at x.

Specifically, it is the function F defined by

the convolution of f with the Gaussian function

The factor 1/√(4π) 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.

Instead of F(x) one also writes W[f](x). Note that F(x) need not exist for every real number x, when the defining integral fails to converge.

The Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient). If the function f 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 t = 1 time units later will be given by the function F. By using values of t different from 1, we can define the generalized Weierstrass transform of f.

The generalized Weierstrass transform provides a means to approximate a given integrable function f arbitrarily well with analytic functions.

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 who studied it extensively. The generalization W_t mentioned below is known in signal analysis as a Gaussian filter and in image processing (when implemented on R²) as a Gaussian blur.

...
Wikipedia