Weierstrass Transform - Transforms of Some Important Functions

Transforms of Some Important Functions

As mentioned above, every constant function is its own Weierstrass transform. The Weierstrass transform of any polynomial is a polynomial of the same degree. Indeed, if H_n denotes the (physicist's) Hermite polynomial of degree n, then the Weierstrass transform of H_n(x/2) is simply xn. This can be shown by exploiting the fact that the generating function for the Hermite polynomials is closely related to the Gaussian kernel used in the definition of the Weierstrass transform.

The Weierstrass transform of the function eax (where a is an arbitrary constant) is ea2 eax. The function eax is thus an eigenvector for the Weierstrass transform. (This is in fact more generally true for all convolution transforms.) By using a=bi where i is the imaginary unit, and using Euler's identity, we see that the Weierstrass transform of the function cos(bx) is e−b2 cos(bx) and the Weierstrass transform of the function sin(bx) is e−b2 sin(bx).

The Weierstrass transform of the function eax2 is if a < 1/4 and undefined if a ≥ 1/4. In particular, by choosing a negative, we see that the Weierstrass transform of a Gaussian function is again a Gaussian function, but a "wider" one.