Weierstrass transform
In mathematics, the Weierstrass transform of a function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } , named after Karl Weierstrass, is a "smoothed" version of f ( x ) {\displaystyle f(x)} obtained by averaging the values of f {\displaystyle f} , weighted with a Gaussian centered at x {\displaystyle x} . Specifically, it is the function F {\displaystyle F} defined by F ( x ) = 1 4 π ∫ − ∞ ∞ f ( y ) e − ( x − y ) 2 4 d y = 1 4 π ∫ − ∞ ∞ f ( x − y ) e − y 2 4 d y , {\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 {\displaystyle f} with the Gaussian function 1 4 π e − x 2 / 4 .