 Weierstrass transform

The Weierstrass transform is Gaussian convolution, $Wf(x)=(4\pi {)}^{-1/2}{\int }_{\mathbb{R}}{e}^{-(x-y{)}^2/4}f(y)dy$. It smooths appropriate functions and is the heat semigroup at unit time. The Brownian transition semigroup uses variance-$t$ Gaussian kernels and therefore supplies the scaled time family.

The Weierstrass transform is a mathematical tool used in the fields of analysis and approximation theory. It is particularly useful in the study of functions and their properties, especially in the context of smoothing and regularization. The Weierstrass transform is named after the German mathematician Karl Weierstrass.