Weierstrass Transform - Generalizations

Generalizations

We can use convolution with the Gaussian kernel (with some t > 0) instead of, thus defining an operator W_t, the generalized Weierstrass transform. For small values of t, W_t is very close to f, but smooth. The larger t, the more this operator averages out and changes f. Physically, W_t corresponds to following the heat (or diffusion) equation for t time units, and this is additive: corresponding to "diffusing for t time units, then s time units, is equivalent to diffusing for s + t time units". One can extend this to t = 0 by setting W₀ to be the identity operator (i.e. convolution with the Dirac delta function), and these then form a one-parameter semigroup of operators.

The kernel used for the generalized Weierstrass transform is sometimes called the Gauss–Weierstrass kernel, and is Green's function for the diffusion equation on R.

W_t can be computed from W: given a function f(x), define a new function f_t(x) = f(x√t); then W_t(x) = W(x/√t), a consequence of the substitution rule.

The Weierstrass transform can also be defined for certain classes of distributions or "generalized functions". For example, the Weierstrass transform of the Dirac delta is the Gaussian . In this context, rigorous inversion formulas can be proved, e.g.

where x₀ is any fixed real number for which F(x₀) exists, the integral extends over the vertical line in the complex plane with real part x₀, and the limit is to be taken in the sense of distributions.

Furthermore, the Weierstrass transform can be defined for real- (or complex-) valued functions (or distributions) defined on Rn. We use the same convolution formula as above but interpret the integral as extending over all of Rn and the expression (x − y)2 as the square of the Euclidean length of the vector x − y; the factor in front of the integral has to be adjusted so that the Gaussian will have a total integral of 1.

More generally, the Weierstrass transform can be defined on any Riemannian manifold: the heat equation can be formulated there (using the manifold's Laplace–Beltrami operator), and the Weierstrass transform W is then given by following the solution of the heat equation for one time unit, starting with the initial "temperature distribution" f.