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 the convolution of f with the Gaussian function 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 Wf. 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. Weierstrass used this transform in his original proof of the Weierstrass approximation theorem. It is also known as the Gauss transform or Gauss–Weierstrass transform after Carl Friedrich Gauss and as the Hille transform after Einar Carl Hille who studied it extensively. The generalization Wt mentioned below is known in signal analysis as a Gaussian filter and in (when implemented on R2) as a Gaussian blur. As mentioned above, every constant function is its own Weierstrass transform. The Weierstrass transform of any polynomial is a polynomial of the same degree, and in fact same leading coefficient (the asymptotic growth is unchanged). Indeed, if Hn denotes the (physicist's) Hermite polynomial of degree n, then the Weierstrass transform of Hn(x/2) is simply xn.