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{{Use American English|date = February 2019}}
{{Short description|"Smoothing" integral transform}}
In [[mathematics]], the '''Weierstrass transform'''<ref>Ahmed I. Zayed, ''Handbook of Function and Generalized Function Transformations'', Chapter 18. CRC Press, 1996.</ref> of a [[function (mathematics)|function]] <math>f : \mathbb{R}\to \mathbb{R}</math>, named after [[Karl Weierstrass]], is a "smoothed" version of <math>f(x)</math> obtained by averaging the values of <math>f</math>, weighted with a Gaussian centered at <math>x</math>.

[[Image:Weierstrass transform.svg|thumb|right|400px|The graph of a function <math>f(x)</math> (black) and its generalized Weierstrass transforms for five <math>t</math> parameters. The standard Weierstrass transform <math>F(x)</math> is given by the case <math>t=1</math> (in green)]]

Specifically, it is the function <math>F</math> defined by
:<math>F(x)=\frac{1}{\sqrt{4\pi}}\int_{-\infty}^\infty f(y) \; e^{-\frac{(x-y)^2}{4}} \; dy = \frac{1}{\sqrt{4\pi}}\int_{-\infty}^\infty f(x-y) \; e^{-\frac{y^2}{4}} \; dy~,</math>
the [[convolution]] of <math>f</math> with the [[Gaussian function]] 
:<math>\frac{1}{\sqrt{4\pi}} e^{-x^2/4}~.</math> 
The factor <math>\frac{1}{\sqrt{4\pi}}</math> 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.

Instead of <math>F(x)</math>  one also writes <math>W[f](x)</math>. Note that <math>F(x)</math> need not exist for every real number <math>x</math>, when the defining integral fails to converge.

The Weierstrass transform is intimately related to the [[heat equation]] (or, equivalently, the [[diffusion equation]] with constant diffusion coefficient). If the function <math>f</math> 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 <math>t=1</math> time units later will be given by the function <math>F</math>. By using values of <math>t</math> different from 1, we can define the '''generalized Weierstrass transform''' of <math>f</math>.

The generalized Weierstrass transform provides a means to approximate a given integrable function <math>f</math> arbitrarily well with [[analytic function]]s.

==Names==
Weierstrass used this transform 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 <math>W_t</math> mentioned below is known in [[signal analysis]] as a [[Gaussian filter]] and in [[image processing]] (when implemented on <math>\mathbb{R}^2</math>) as a [[Gaussian blur]].

==Transforms of some important functions==

=== Constant Functions ===
Every [[constant function]] is its own Weierstrass transform. 

=== Polynomials ===
The Weierstrass transform of any [[polynomial]] is a polynomial of the same degree, and in fact has the same leading coefficient (the [[asymptotic growth]] is unchanged). 
Indeed, if <math>H_n</math> denotes the [[Hermite polynomials|(physicist's) Hermite polynomial]] of degree <math>n</math>, then the Weierstrass transform of <math>H_n(x/2)</math> is simply <math>x^n</math>. 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.

=== Exponentials, Sines, and Cosines  ===
The Weierstrass transform of the exponential function <math>x \mapsto \exp(ax)</math> (where <math>a</math> is an arbitrary constant) is <math>x \mapsto \exp(a^2)\exp(ax)</math>. The function  <math>\exp(ax)</math> is thus an [[eigenfunction]] of the Weierstrass transform, with eigenvalue <math>\exp(a^2)</math>.{{NoteTag|More generally, <math>x \mapsto \exp(ax)</math> is an eigenfunction for ''any'' convolution transforms.}}

Using Weierstrass transform of <math>x \mapsto \exp(ax)</math> with <math>a =bi</math> where <math>b</math> is an arbitrary real constant and <math>i</math> is the [[imaginary unit]], and applying [[Euler's identity]], one sees that the Weierstrass transform of the function <math>\cos(bx)</math> is <math>\exp(-b^2)\cos(bx)</math> and the Weierstrass transform of the function <math>\sin(bx)</math> is <math> \exp(-b^2) \sin(bx)</math>.

=== Gaussian Functions  ===
The Weierstrass transform of the function <math>x \mapsto \exp(ax^2)</math> is 
<math display="block">
F(x) = \begin{cases}
  \dfrac{1}{\sqrt{1-4a}} \exp\left(\dfrac{ax^2}{1-4a}\right) & \text{if } a < 1/4 \\[5pt]
  \text{undefined} & \text{if } a \geq 1/4.
\end{cases}
</math> 
Of particular note is when <math>a</math> is chosen to be negative. If <math>a < 0</math>, then <math>\exp(ax^2)</math> is a Gaussian function and its Weierstrass transform is also a Gaussian function, but a "wider" one.

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

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

If <math>f</math> is integrable over the whole real axis (i.e. <math>f \in L^1(\R)</math>), then so is its Weierstrass transform <math>F</math>, and if furthermore <math>f(x)\ge 0</math> for all <math>x</math>, then also <math>F(x)\ge 0</math> for all <math>x</math> and the integrals of <math>f</math> and <math>F</math> 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 <math>1 \le p \le +\infty</math> and <math>f \in L^p(\R)</math>, we have <math>F \in L^p(\R)</math> and <math>||F||_p \le ||f||_p</math>. The Weierstrass transform consequently yields a [[bounded operator]] <math>W : L^p(\R)\to L^p(\R)</math>.

If <math>f</math> is sufficiently smooth, then the Weierstrass transform of the <math>k</math>-th [[derivative]] of <math>f</math> is equal to the <math>k</math>-th derivative of the Weierstrass transform of <math>f</math>.

There is a formula relating the Weierstrass transform ''W'' and the [[two-sided Laplace transform]] <math>L</math>. If we define

:<math>g(x)=e^{-\frac{x^2}{4}} f(x)</math>

then

:<math>W[f](x)=\frac{1}{\sqrt{4\pi}} e^{-x^2/4} L[g]\left(-\frac{x}{2}\right).</math>

=== Low-pass filter ===
{{details|Gaussian blur#Low-pass filter}}
We have seen above that the Weierstrass transform of <math>\cos(bx)</math> is <math>e^{-b^2}\cos(bx)</math>, and analogously for <math>\sin(bx)</math>. In terms of [[signal analysis]], this suggests that if the signal <math>f</math> contains the frequency <math>b</math> (i.e. contains a summand which is a combination of <math>\sin(bx)</math> and <math>\cos(bx)</math>), then the transformed signal <math>F</math> will contain the same frequency, but with an [[amplitude]] multiplied by the factor <math>e^{-b^2}</math>. This has the consequence that higher frequencies are reduced more than lower ones, and the Weierstrass transform thus acts as a [[low-pass filter]]. 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 ''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, which is another way of describing the "smoothing" property of the Weierstrass transform.

==The inverse transform==
The following formula, closely related to the [[Laplace transform]] of a Gaussian function, and a real analogue to the [[Hubbard–Stratonovich transformation]], is relatively easy to establish:

:<math>e^{u^2}=\frac{1}{\sqrt{4\pi}} \int_{-\infty}^\infty e^{-uy} e^{-y^2/4}\;dy.</math>

Now replace ''u'' with the formal differentiation operator ''D''&nbsp;=&nbsp;''d''/''dx'' and utilize the Lagrange [[shift operator]]  
:<math>e^{-yD}f(x)=f(x-y)</math>, 
(a consequence of the [[Taylor series]] formula and the definition of the [[exponential function]]), to obtain
:<math>
\begin{align}
e^{D^2}f(x) & = \frac{1}{\sqrt{4\pi}} \int_{-\infty}^\infty e^{-yD}f(x) e^{-y^2/4}\;dy \\
& =\frac{1}{\sqrt{4\pi}} \int_{-\infty}^\infty f(x-y) e^{-y^2/4}\;dy=W[f](x)
\end{align}
</math>
to thus obtain the following formal expression for the Weierstrass transform <math>W</math>,
{{Equation box 1
|indent =:
|equation = <math>W=e^{D^2} ~,</math> 
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
where the operator on the right is to be understood as acting on the function ''f''(''x'')  as
:<math>e^{D^2} f(x) = \sum_{k=0}^\infty \frac{D^{2k}f(x)}{k!}~.</math>

The above formal derivation glosses over details of convergence, and the formula <math>W = e^{D^2}</math> is thus not universally valid; there are several functions <math>f</math> which have a well-defined Weierstrass transform, but for which <math>e^{D^2}(f)</math> 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
:<math>W^{-1}=e^{-D^2}  ~. </math>
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.<ref>G. G. Bilodeau, "[http://projecteuclid.org/euclid.dmj/1077470134 The Weierstrass Transform and Hermite Polynomials]". ''Duke Mathematical Journal'' 29 (1962), p. 293-308</ref>

One may, alternatively,  attempt to invert the Weierstrass transform in a slightly different way: given the analytic function
:<math>F(x)=\sum_{n=0}^\infty a_n x^n  ~,</math>
apply <math>W^{-1}</math> to obtain
:<math>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)</math>
once more using a fundamental property of the (physicists') [[Hermite polynomials]] <math>H_n</math>.

Again, this formula for <math>f(x)</math> is at best formal, since one didn't check whether the final series converges. But if, for instance, <math>f \in L^2(\R)</math>, then knowledge of all the derivatives of <math>F</math> at <math>x=0</math> suffices to yield the coefficients <math>a_n</math>; and to thus reconstruct <math>f</math> as a series of [[Hermite polynomials]].

A third method of inverting 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==
{{see also|Heat equation # Fundamental solutions|Heat kernel}}

We can use convolution with the Gaussian kernel <math display="block">\frac{1}{\sqrt{4\pi t}} e^{-\frac{x^2}{4t}}, \quad t > 0,</math> instead of <math display="block">\frac{1}{\sqrt{4\pi}} e^{-\frac{x^2}{4}},</math> thus defining an operator {{math|''W<sub>t</sub>''}}, the generalized Weierstrass transform.

For small values of <math>t</math>, <math>W_t[f]</math> is very close to <math>f</math>, but smooth. The larger <math>t</math>, the more this operator averages out and changes <math>f</math>. Physically, <math>W_t</math> corresponds to following the heat (or diffusion) equation for <math>t</math> time units, and this is additive, 
<math display="block">W_s \circ W_t = W_{s+t},</math> 
corresponding to "diffusing for <math>t</math> time units, then <math>s</math> time units, is equivalent to diffusing for <math>s+t</math> time units". One can extend this to <math>t=0</math> by setting <math>W_0</math> 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 <math display="block">\frac{1}{\sqrt{4\pi t}} e^{-\frac{x^2}{4t}},</math> used for the generalized Weierstrass transform is sometimes called the '''Gauss–Weierstrass kernel''', and is [[Green's function#Table of Green's functions|Green's function for the diffusion equation]] <math display="block">(\partial_t -D^2) (e^{tD^2} f(x))=0,</math> on <math>\R</math>.
<math>W_t</math> can be computed from <math>W</math>: given a function <math>f(x)</math>, define a new function <math display="block">f_t(x) := f(x\sqrt{t}),</math> then <math display="block">W_t[f](x) = W[f_t](x\sqrt{t}),</math> a consequence of the [[substitution rule]].

The Weierstrass transform can also be defined for certain classes of [[distribution (mathematics)|distribution]]s or "generalized functions".<ref>Yu A. Brychkov, A. P. Prudnikov. ''Integral Transforms of Generalized Functions'', Chapter 5. CRC Press, 1989</ref> For example, the Weierstrass transform of the [[Dirac delta]] is the Gaussian <math display="block">\frac{1}{\sqrt{4\pi}} e^{-x^2/4}.</math>
In this context, rigorous inversion formulas can be proved, e.g.,
<math display="block">f(x)=\lim_{r\to\infty}\frac{1}{i\sqrt{4\pi}} \int_{x_0-ir}^{x_0+ir} F(z)e^{\frac{(x-z)^2}{4}}\;dz,</math>
where <math>x_0</math> is any fixed real number for which <math>F(x_0)</math> exists, the integral extends over the vertical line in the complex plane with real part <math>x_0</math>, 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 <math>\R^n</math>. We use the same convolution formula as above but interpret the integral as extending over all of <math>\R^n</math> and the expression <math>(x-y)^2</math> as the square of the [[Euclidean distance|Euclidean length]] of the vector <math>x-y</math>; 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 <math>W[f]</math> is then given by following the solution of the heat equation for one time unit, starting with the initial "temperature distribution" <math>f</math>.

==Related transforms==
If one considers convolution with the kernel  <math>\frac{1}{(1+x^2)\pi}</math> instead of with a Gaussian, one obtains the [[Poisson kernel|Poisson transform]] which smoothes and averages a given function in a manner similar to the Weierstrass transform.

== See also ==
* [[Gaussian blur]]
* [[Gaussian filter]]
* [[Husimi Q representation]]
* [[Heat equation#Fundamental solutions]]


==Notes==
{{NoteFoot}}

==References==
{{reflist}}

{{DEFAULTSORT:Weierstrass Transform}}
[[Category:Integral transforms]]
[[Category:Mathematical physics]]