Weierstrass transform in complex variable - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T02:37:10Z http://mathoverflow.net/feeds/question/105679 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105679/weierstrass-transform-in-complex-variable Weierstrass transform in complex variable Hu Yi Chen 2012-08-28T03:54:57Z 2012-08-28T03:54:57Z <p>The usual Weierstrass transform of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ is defined as: $$e^{D^2/2}f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-yD}f(x)e^{-y^2/2} dy=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(x-y)e^{-y^2/2}dy$$ where $D=\frac{d}{dx}$.</p> <p>Now if $D$ is with respect to complex variable $z$, how will the Weierstrass transform be different from the one above?</p>