Weierstrass transform

In mathematics, the Weierstrass transform [Ahmed I. Zayed, "Handbook of Function and Generalized Function Transformations", Chapter 18. CRC Press, 1996.] of a function "f" : R → R is the function "F" defined by

:$F(x)=frac\{1\}\{sqrt\{4piint\_\{-infty\}^infty\; f(y)\; ;\; e^\{-frac\{(x-y)^2\}\{4\; ;\; dy\; =\; frac\{1\}\{sqrt\{4piint\_\{-infty\}^infty\; f(x-y)\; ;\; e^\{-frac\{y^2\}\{4\; ;\; dy,$

the convolution of "f" with the Gaussian function $frac\{1\}\{sqrt\{4pi\; e^\{-x^2/4\}$. Instead of "F"("x") we also write "W" ["f"] ("x"). Note that "F"("x") need not exist for every real number "x", because the defining integral may fail to converge.

The Weierstrass transform "F" can be viewed as a "smoothened" version of "f": the value "F"("x") is obtained by averaging the values of "f", weighted with a Gaussian centered at "x". The number 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.

The Weierstrass transform is intimately related to the heat equation (or, what is the same thing, the diffusion equation). 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".

Names

The Weierstrass transform is named after Karl Weierstrass who used it 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 "W_t" mentioned below is known in signal analysis as a Gaussian filter and in image processing (when implemented on R²) as a Gaussian blur.

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 "x"^"n". 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 "e"^"ax" (where "a" is an arbitrary constant) is "e"^"a"2 "e"^"ax". The function "e"^"ax" is thus an eigenvalue for the Weierstrass transform; this is a general fact about 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"^-"b"2 cos("bx") and theWeierstrass transform of the function sin("bx") is"e"^-"b"2 sin("bx").

The Weierstrass transform of the function $e^\{ax^2\}$ is $frac\{1\}\{sqrt\{1-4ae^\{frac\{ax^2\}\{sqrt\{1-4a\}$ if "a"<1/4 and undefined if "a"&ge;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.

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 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 the function is analytic there; moreover, "F"("x") will exist for all complex values of "x" with "a" &le; Re("x") &le; "b", and the function is holomorphic on that strip. This is the formal statement of the "smoothness" of "F" mentioned above.

If "f" is integrable over the whole real axis (i.e. "f"&isin;L¹(R)), then so is its Weierstrass transform "F", and if furthermore "f"("x") &ge; 0 for all "x", then also "F"("x") &ge; 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.

Using the above, one can show that for 0<"p"&le;&infin; and "f"&isin;L^p(R), we have "F"&isin;L^p(R) and ||"F"||_p &le; ||"f"||_p. The Weierstrass transform consequently yields a bounded operator W : L^p(R) &rarr; L^p(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".

We have seen above that the Weierstrass transform of cos("bx") is "e"^-"b"2 cos("bx"), and analogously for sin("bx"). In terms of signal analysis, this suggests that if the signal "f" contains the frequency "b" (i.e. contains a summand which is a combination of sin("bx") and cos("bx")), then the transformed signal "F" will contain the same frequency, but with an amplitude reduced by the factor "e"^-"b"2. This has the consequence that higher frequencies are reduced more than lower frequencies. This can also be shown with the continuous Fourier transform, as follows. The Fourier transform analyzes a signal in terms of its frequencies, transforms convolutions into products, and transforms Gaussians into Gaussians. The Weierstrass transform is convolution with a Gaussian and is therefore nothing but "multiplication" of the Fourier transformed signal with a Gaussian, followed by application of the inverse Fourier transform. This multiplication with a Gaussian in frequency space blends out high frequencies, with is another way of describing the "smoothing" property of the Weierstrass transform.

There is a formula relating the Weierstrass transform "W" and the two-sided Laplace transform "L". If we define :$g(x)=e^\{-frac\{x^2\}\{4\; f(x)$then:$W\; [f]\; (x)=frac\{1\}\{sqrt\{4pi\; e^\{-x^2/4\}\; L\; [g]\; left(-frac\{x\}\{2\}\; ight)$

The inverse

The following formula, closely related to the Laplace transform of a Gaussian function, is relatively easy to establish::$e^\{u^2\}=frac\{1\}\{sqrt\{4pi\; int\_\{-infty\}^\{infty\}\; e^\{-uy\}\; e^\{-y^2/4\};dy$Now replace "u" with the formal differentiation operator $D=frac\{d\}\{dx\}$ and use the fact that formally $e^\{-yD\}f(x)=f(x-y)$, a consequence of the Taylor series formula and the definition of the exponential function.:$e^\{D^2\}f(x)=frac\{1\}\{sqrt\{4pi\; int\_\{-infty\}^\{infty\}\; e^\{-yD\}f(x)\; e^\{-y^2/4\};dy\; =frac\{1\}\{sqrt\{4pi\; int\_\{-infty\}^\{infty\}\; f(x-y)\; e^\{-y^2/4\};dy=W\; [f]\; (x)$and we obtain the following formal expression for the Weierstrass transform "W"::$W=e^\{D^2\}$where the operator on the right is to be understood as:$e^\{D^2\}\; f(x)\; =\; sum\_\{k=0\}^infty\; frac\{D^\{2k\}f(x)\}\{k!\}.$The derivation above glosses over many details of convergence, and the formula "W"="e"^"D"2 is therefore not universally valid; there are many functions "f" which have a well-defined Weierstrass transform but for which "e"^"D"2"f"("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 :$W^\{-1\}=e^\{-D^2\}.$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. [G. G. Bilodeau, "The Weierstrass Transform and Hermide Polynomials". "Duke Mathematical Journal" 29 (1962), p. 293-308]

We can also attempt to invert the Weierstrass transform in a different way: given the analytic function:$F(x)=sum\_\{n=0\}^infty\; a\_n\; x^n$we apply "W"^-1 to obtain:$f(x)=W^\{-1\}\; [F(x)]\; =sum\_\{n=0\}^infty\; a\_n\; W^\{-1\}\; [x^n]\; =sum\_\{n=0\}^infty\; a\_n\; H\_n(x/2)$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"&isin;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.

Generalizations and related transforms

We can use convolution with the Gaussian kernel $frac\{1\}\{sqrt\{4pi\; t\; e^\{-frac\{x^2\}\{4t$ (with some "t">0) instead of $frac\{1\}\{sqrt\{4pi\; e^\{-frac\{x^2\}\{4$, thus defining an operator "W_t". For small values of "t", "W_t" ["f"] 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 equation for "t" time units. "W_t" can be computed from "W": given a function "f"("x"), define a new function "f"_"t"("x") = "f"("x"&radic;"t"); then "W_t" ["f"] ("x") = "W" ["f_t"] ("x"/&radic;"t"), a consequence of the substitution rule.

The Weierstrass transform can also be defined for certain classes of distributions or "generalized functions". [Yu A. Brychkov, A. P. Prudnikov. "Integral Transforms of Generalized Functions", Chapter 5. CRC Press, 1989] . For example, the Weierstrass transform of the Dirac delta is the Gaussian $frac\{1\}\{sqrt\{4pi\; e^\{-x^2/4\}$. In this context, rigorous inversion formulas can be proved, e.g.:$f(x)=lim\_\{r\; oinfty\}frac\{1\}\{isqrt\{4pi\; int\_\{x\_0-ir\}^\{x\_0+ir\}\; F(z)e^\{frac\{(x-z)^2\}\{4;dz$where "x"₀ is any fixed real number for which "F"("x"₀) exists, the integral extends over the vertical line in the complex plane with real part "x"₀, 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 R^"n". We use the same convolution formula as above but interpret the integral as extending over all of R^"n" 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 defined there, and the Weierstrass transform "W" ["f"] is then given by following the heat equation for one time unit, starting from the initial "temperature distribution" "f".

If one considers convolution with the kernel $frac\{1\}\{1+x^2\}$ instead of with a Gaussian, one obtains the Poisson transform which smoothes and averages a given function in a manner similar to the Weierstrass transform.

References