Weierstrass transform

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 $W\_t$ mentioned below is known in signal analysis as a Gaussian filter and in image processing (when implemented on $\backslash mathbb\{R\}^2$) as a Gaussian blur.

Every constant function is its own Weierstrass transform.

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 $H\_n$ 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 exponential function $x\; \backslash mapsto\; \backslash exp(ax)$ (where $a$ is an arbitrary constant) is $x\; \backslash mapsto\; \backslash exp(a^2)\backslash exp(ax)$. The function $\backslash exp(ax)$ is thus an eigenfunction of the Weierstrass transform, with eigenvalue $\backslash exp(a^2)$.{{NoteTag|More generally, $x\; \backslash mapsto\; \backslash exp(ax)$ is an eigenfunction for any convolution transforms.}}

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

The Weierstrass transform of the function $x\; \backslash mapsto\; \backslash exp(ax^2)$ is

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}

Of particular note is when $a$ is chosen to be negative. If , then $\backslash exp(ax^2)$ is a Gaussian function and its Weierstrass transform is also a Gaussian function, but a "wider" one.

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 $a\; \backslash le\; \backslash Re(x)\backslash le\; b$ 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. $f\; \backslash in\; L^1(\backslash R)$), then so is its Weierstrass transform $F$, and if furthermore $f(x)\backslash ge\; 0$ for all $x$, then also $F(x)\backslash ge\; 0$ 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, or that the total amount of diffusing material is conserved by the diffusion equation.

Using the above, one can show that for $1\; \backslash le\; p\; \backslash le\; +\backslash infty$ and $f\; \backslash in\; L^p(\backslash R)$, we have $F\; \backslash in\; L^p(\backslash R)$ and $||F||\_p\; \backslash le\; ||f||\_p$. The Weierstrass transform consequently yields a bounded operator $W\; :\; L^p(\backslash R)\backslash to\; L^p(\backslash R)$.

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

:$g(x)=e^\{-\backslash frac\{x^2\}\{4\}\}\; f(x)$

then

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

{{details|Gaussian blur#Low-pass filter}}

We have seen above that the Weierstrass transform of $\backslash cos(bx)$ is $e^\{-b^2\}\backslash cos(bx)$, and analogously for $\backslash sin(bx)$. 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 $\backslash sin(bx)$ and $\backslash cos(bx)$), then the transformed signal $F$ will contain the same frequency, but with an amplitude multiplied by the factor $e^\{-b^2\}$. 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 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:

:$e^\{u^2\}=\backslash frac\{1\}\{\backslash sqrt\{4\backslash pi\}\}\; \backslash int\_\{-\backslash infty\}^\backslash infty\; e^\{-uy\}\; e^\{-y^2/4\}\backslash ;dy.$

Now replace u with the formal differentiation operator D = d/dx and utilize the Lagrange shift operator

:$e^\{-yD\}f(x)=f(x-y)$,

(a consequence of the Taylor series formula and the definition of the exponential function), to obtain

:$$

\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}

to thus obtain the following formal expression for the Weierstrass transform $W$,

{{Equation box 1

|indent =:

|equation = $W=e^\{D^2\}\; ~,$

|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

:$e^\{D^2\}\; f(x)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{D^\{2k\}f(x)\}\{k!\}~.$

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

One may, alternatively, attempt to invert the Weierstrass transform in a slightly different way: given the analytic function

:$F(x)=\backslash sum\_\{n=0\}^\backslash infty\; a\_n\; x^n\; ~,$

apply $W^\{-1\}$ to obtain

:$f(x)=W^\{-1\}[F(x)]=\backslash sum\_\{n=0\}^\backslash infty\; a\_n\; W^\{-1\}[x^n]=\backslash sum\_\{n=0\}^\backslash infty\; a\_n\; H\_n(x/2)$

once more using a fundamental property of the (physicists') Hermite polynomials $H\_n$.

Again, this formula for $f(x)$ is at best formal, since one didn't check whether the final series converges. But if, for instance, $f\; \backslash in\; L^2(\backslash R)$, then knowledge of all the derivatives of $F$ at $x=0$ suffices to yield the coefficients $a\_n$; and to thus reconstruct $f$ 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.

{{see also|Heat equation # Fundamental solutions|Heat kernel}}

We can use convolution with the Gaussian kernel $$\backslash frac\{1\}\{\backslash sqrt\{4\backslash pi\; t\}\}\; e^\{-\backslash frac\{x^2\}\{4t\}\},\; \backslash quad\; t\; >\; 0,$$ instead of $$\backslash frac\{1\}\{\backslash sqrt\{4\backslash pi\}\}\; e^\{-\backslash frac\{x^2\}\{4\}\},$$ thus defining an operator {{math|W_t}}, the generalized Weierstrass transform.

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 (or diffusion) equation for $t$ time units, and this is additive,

$$W\_s\; \backslash circ\; W\_t\; =\; W\_\{s+t\},$$

corresponding to "diffusing for $t$ time units, then $s$ time units, is equivalent to diffusing for $s+t$ time units". One can extend this to $t=0$ by setting $W\_0$ 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 $$\backslash frac\{1\}\{\backslash sqrt\{4\backslash pi\; t\}\}\; e^\{-\backslash frac\{x^2\}\{4t\}\},$$ used for the generalized Weierstrass transform is sometimes called the Gauss–Weierstrass kernel, and is Green's function for the diffusion equation $$(\backslash partial\_t\; -D^2)\; (e^\{tD^2\}\; f(x))=0,$$ on $\backslash R$.

$W\_t$ can be computed from $W$: given a function $f(x)$, define a new function $$f\_t(x)\; :=\; f(x\backslash sqrt\{t\}),$$ then $$W\_t[f](x)\; =\; W[f\_t](x\backslash sqrt\{t\}),$$ 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 $$\backslash frac\{1\}\{\backslash sqrt\{4\backslash pi\}\}\; e^\{-x^2/4\}.$$

In this context, rigorous inversion formulas can be proved, e.g.,

$$f(x)=\backslash lim\_\{r\backslash to\backslash infty\}\backslash frac\{1\}\{i\backslash sqrt\{4\backslash pi\}\}\; \backslash int\_\{x\_0-ir\}^\{x\_0+ir\}\; F(z)e^\{\backslash frac\{(x-z)^2\}\{4\}\}\backslash ;dz,$$

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 $\backslash R^n$. We use the same convolution formula as above but interpret the integral as extending over all of $\backslash R^n$ 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 formulated there (using the manifold's Laplace–Beltrami operator), and the Weierstrass transform $W[f]$ is then given by following the solution of the heat equation for one time unit, starting with the initial "temperature distribution" $f$.

If one considers convolution with the kernel $\backslash frac\{1\}\{(1+x^2)\backslash pi\}$ 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.

• Gaussian blur
• Gaussian filter
• Husimi Q representation
• Heat equation#Fundamental solutions

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Category:Integral transforms

Category:Mathematical physics