Weierstrass Transform - General Properties

General Properties

The Weierstrass transform assigns to each function f a new function F; this assignment is linear. It is also translation-invariant, meaning that the transform of the function f(x + a) is F(x + a). Both of these facts are more generally true for any integral transform defined via convolution.

If the transform F(x) exists for the real numbers x = a and x = b, then it also exists for all real values in between and forms an analytic function there; moreover, F(x) will exist for all complex values of x with a ≤ Re(x) ≤ b and forms a holomorphic function on that strip of the complex plane. This is the formal statement of the "smoothness" of F mentioned above.

If f is integrable over the whole real axis (i.e. f ∈ L1(R)), then so is its Weierstrass transform F, and if furthermore f(x) ≥ 0 for all x, then also F(x) ≥ 0 for all x and the integrals of f and F are equal. This expresses the physical fact that the total thermal energy or heat is conserved by the heat equation, or that the total amount of diffusing material is conserved by the diffusion equation.

Using the above, one can show that for 0 < p ≤ ∞ and f ∈ Lp(R), we have F ∈ Lp(R) and ||F||_p ≤ ||f||_p. The Weierstrass transform consequently yields a bounded operator W : Lp(R) → Lp(R).

If f is sufficiently smooth, then the Weierstrass transform of the k-th derivative of f is equal to the k-th derivative of the Weierstrass transform of f.

There is a formula relating the Weierstrass transform W and the two-sided Laplace transform L. If we define

then