Weierstrass Transform - The Inverse

The Inverse

The following formula, closely related to the Laplace transform of a Gaussian function, is relatively easy to establish:

Now replace u with the formal differentiation operator D = d/dx and use the fact that formally, a consequence of the Taylor series formula and the definition of the exponential function.

and we obtain the following formal expression for the Weierstrass transform W:

where the operator on the right is to be understood as acting on the function f(x) via

The derivation above glosses over many details of convergence, and the formula W = eD2 is therefore not universally valid; there are many functions f which have a well-defined Weierstrass transform but for which eD2f(x) cannot be meaningfully defined. Nevertheless, the rule is still quite useful and can for example be used to derive the Weierstrass transforms of polynomials, exponential and trigonometric functions mentioned above.

The formal inverse of the Weierstrass transform is thus given by

Again this formula is not universally valid but can serve as a guide. It can be shown to be correct for certain classes of functions if the right-hand side operator is properly defined.

We can also attempt to invert the Weierstrass transform in a different way: given the analytic function

we apply W−1 to obtain

once more using the (physicist's) Hermite polynomials H_n. Again, this formula for f(x) is at best formal since we didn't check whether the final series converges. But if for instance f ∈ L2(R), then knowledge of all the derivatives of F at x = 0 is enough to find the coefficients a_n and reconstruct f as a series of Hermite polynomials.

A third method to invert the Weierstrass transform exploits its connection to the Laplace transform mentioned above, and the well-known inversion formula for the Laplace transform. The result is stated below for distributions.