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

${\displaystyle F(x)={\frac {1}{\sqrt {4\pi }}}\int _{-\infty }^{\infty }f(y)\;e^{-{\frac {(x-y)^{2}}{4}}}\;dy={\frac {1}{\sqrt {4\pi }}}\int _{-\infty }^{\infty }f(x-y)\;e^{-{\frac {y^{2}}{4}}}\;dy~,}$

the convolution of f with the Gaussian function

${\displaystyle {\frac {1}{\sqrt {4\pi }}}e^{-x^{2}/4}~.}$

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.