Weierstrass transform

In mathematics, the Weierstrass transform^[1] of a function ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$, named after Karl Weierstrass, is a "smoothed" version of ${\displaystyle f(x)}$ obtained by averaging the values of ${\displaystyle f}$, weighted with a Gaussian centered at ${\displaystyle x}$.

Specifically, it is the function ${\displaystyle F}$ 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 ${\displaystyle f}$ with the Gaussian function

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

The factor ${\displaystyle {\frac {1}{\sqrt {4\pi }}}}$ 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 ${\displaystyle F(x)}$ one also writes ${\displaystyle W[f](x)}$. Note that ${\displaystyle F(x)}$ need not exist for every real number ${\displaystyle x}$, 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 ${\displaystyle f}$ 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 ${\displaystyle t=1}$ time units later will be given by the function ${\displaystyle F}$. By using values of ${\displaystyle t}$ different from 1, we can define the generalized Weierstrass transform of ${\displaystyle f}$.

The generalized Weierstrass transform provides a means to approximate a given integrable function ${\displaystyle f}$ arbitrarily well with analytic functions.