Weierstrass Transform
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Weierstrass transform of a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
f : \mathbb\to \mathbb, named after
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...
, is a "smoothed" version of f(x) obtained by averaging the values of f, weighted with a Gaussian centered at x. Specifically, it is the function F defined by :F(x)=\frac\int_^\infty f(y) \; e^ \; dy = \frac\int_^\infty f(x-y) \; e^ \; dy~, the
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of f with the
Gaussian function In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function (mathematics), function of the base form f(x) = \exp (-x^2) and with parametric extension f(x) = a \exp\left( -\frac \right) for arbitrary real number, rea ...
:\frac e^~. The factor \frac 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 F(x) one also writes W x). Note that F(x) need not exist for every real number x, when the defining integral fails to converge. The Weierstrass transform is intimately related to the
heat equation In mathematics and physics (more specifically thermodynamics), the heat equation is a parabolic partial differential equation. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quanti ...
(or, equivalently, the
diffusion equation The diffusion equation is a parabolic partial differential equation. In physics, it describes the macroscopic behavior of many micro-particles in Brownian motion, resulting from the random movements and collisions of the particles (see Fick's l ...
with constant diffusion coefficient). If the function f describes the initial temperature at each point of an infinitely long rod that has constant
thermal conductivity The thermal conductivity of a material is a measure of its ability to heat conduction, conduct heat. It is commonly denoted by k, \lambda, or \kappa and is measured in W·m−1·K−1. Heat transfer occurs at a lower rate in materials of low ...
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. The generalized Weierstrass transform provides a means to approximate a given integrable function f arbitrarily well with
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s.


Names

Weierstrass used this transform in his original proof of the
Weierstrass approximation theorem Karl Theodor Wilhelm Weierstrass (; ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the " father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school t ...
. It is also known as the Gauss transform or Gauss–Weierstrass transform after
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observatory and ...
and as the Hille transform after Einar Carl Hille who studied it extensively. The generalization W_t mentioned below is known in
signal analysis Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, potential fields, seismic signals, altimetry processing, and scientific measurements. Signa ...
as a
Gaussian filter In electronics and signal processing, mainly in digital signal processing, a Gaussian filter is a filter (signal processing), filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response would h ...
and in
image processing An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a pr ...
(when implemented on \mathbb^2) as a
Gaussian blur In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss). It is a widely used effect in graphics software, ...
.


Transforms of some important functions


Constant Functions

Every
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. Basic properties As a real-valued function of a real-valued argument, a constant function has the general form or just For example, ...
is its own Weierstrass transform.


Polynomials

The Weierstrass transform of any
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
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 In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression invo ...
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 x \mapsto \exp(ax) (where a is an arbitrary constant) is x \mapsto \exp(a^2)\exp(ax). The function \exp(ax) is thus an
eigenfunction In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, th ...
of the Weierstrass transform, with eigenvalue \exp(a^2). Using Weierstrass transform of x \mapsto \exp(ax) with a =bi where b is an arbitrary real constant and i is the
imaginary unit The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
, and applying
Euler's identity In mathematics, Euler's identity (also known as Euler's equation) is the Equality (mathematics), equality e^ + 1 = 0 where :e is E (mathematical constant), Euler's number, the base of natural logarithms, :i is the imaginary unit, which by definit ...
, one sees that the Weierstrass transform of the function \cos(bx) is \exp(-b^2)\cos(bx) and the Weierstrass transform of the function \sin(bx) is \exp(-b^2) \sin(bx).


Gaussian Functions

The Weierstrass transform of the function x \mapsto \exp(ax^2) is F(x) = \begin \dfrac \exp\left(\dfrac\right) & \text a < 1/4 \\ pt \text & \text a \geq 1/4. \end Of particular note is when a is chosen to be negative. If a < 0, then \exp(ax^2) 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 f a new function F; this assignment is
linear In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a '' function'' (or '' mapping''); * linearity of a '' polynomial''. An example of a linear function is the function defined by f(x) ...
. 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 In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
there; moreover, F(x) will exist for all
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
values of x with a \le \Re(x)\le b and forms a
holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
on that strip of the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
. This is the formal statement of the "smoothness" of F mentioned above. If f is integrable over the whole real axis (i.e. f \in L^1(\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 In thermodynamics, heat is energy in transfer between a thermodynamic system and its surroundings by such mechanisms as thermal conduction, electromagnetic radiation, and friction, which are microscopic in nature, involving sub-atomic, ato ...
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 \le p \le +\infty and f \in L^p(\R), we have F \in L^p(\R) and , , F, , _p \le , , f, , _p. The Weierstrass transform consequently yields a
bounded operator In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector ...
W : L^p(\R)\to L^p(\R). If f is sufficiently smooth, then the Weierstrass transform of the k-th
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
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 In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment-generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Melli ...
L. If we define :g(x)=e^ f(x) then :W x)=\frac e^ L left(-\frac\right).


Low-pass filter

We have seen above that the Weierstrass transform of \cos(bx) is e^\cos(bx), and analogously for \sin(bx). In terms of
signal analysis Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, potential fields, seismic signals, altimetry processing, and scientific measurements. Signa ...
, 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 The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
multiplied by the factor e^. This has the consequence that higher frequencies are reduced more than lower ones, and the Weierstrass transform thus acts as a
low-pass filter A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filt ...
. This can also be shown with the
continuous Fourier transform In mathematics, the Fourier transform (FT) is an integral transform that takes a function (mathematics), function as input then outputs another function that describes the extent to which various Frequency, frequencies are present in the origin ...
, 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 In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a f ...
of a Gaussian function, and a real analogue to the
Hubbard–Stratonovich transformation The Hubbard–Stratonovich (HS) transformation is an exact mathematical transformation invented by Russian physicist Ruslan L. Stratonovich and popularized by British physicist John Hubbard. It is used to convert a particle theory into its resp ...
, is relatively easy to establish: :e^=\frac \int_^\infty e^ e^\;dy. Now replace ''u'' with the formal differentiation operator ''D'' = ''d''/''dx'' and utilize the Lagrange
shift operator In mathematics, and in particular functional analysis, the shift operator, also known as the translation operator, is an operator that takes a function to its translation . In time series analysis, the shift operator is called the '' lag opera ...
:e^f(x)=f(x-y), (a consequence of the
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
formula and the definition of the exponential function), to obtain : \begin e^f(x) & = \frac \int_^\infty e^f(x) e^\;dy \\ & =\frac \int_^\infty f(x-y) e^\;dy=W x) \end to thus obtain the following formal expression for the Weierstrass transform W, where the operator on the right is to be understood as acting on the function ''f''(''x'') as :e^ f(x) = \sum_^\infty \frac~. The above formal derivation glosses over details of convergence, and the formula W = e^ is thus not universally valid; there are several functions f which have a well-defined Weierstrass transform, but for which e^(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^=e^ ~. 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. One may, alternatively, attempt to invert the Weierstrass transform in a slightly different way: given the analytic function :F(x)=\sum_^\infty a_n x^n ~, apply W^ to obtain :f(x)=W^ (x)\sum_^\infty a_n W^ ^n\sum_^\infty a_n H_n(x/2) once more using a fundamental property of the (physicists')
Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...
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 \in L^2(\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 In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well a ...
. 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

We can use convolution with the Gaussian kernel \frac e^, \quad t > 0, instead of \frac e^, thus defining an operator , the generalized Weierstrass transform. For small values of t, W_t /math> 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 \circ W_t = W_, 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 In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
), and these then form a one-parameter semigroup of operators. The kernel \frac e^, used for the generalized Weierstrass transform is sometimes called the Gauss–Weierstrass kernel, and is Green's function for the diffusion equation (\partial_t -D^2) (e^ f(x))=0, on \R. W_t can be computed from W: given a function f(x), define a new function f_t(x) := f(x\sqrt), then W_t x) = W _tx\sqrt), a consequence of the substitution rule. The Weierstrass transform can also be defined for certain classes of
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...
s 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 In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
is the Gaussian \frac e^. In this context, rigorous inversion formulas can be proved, e.g., f(x)=\lim_\frac \int_^ F(z)e^\;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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' ...
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 In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
: the heat equation can be formulated there (using the manifold's
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named aft ...
), and the Weierstrass transform W /math> is then given by following the solution of the heat equation for one time unit, starting with the initial "temperature distribution" f.


Related transforms

If one considers convolution with the kernel \frac 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.


See also

*
Gaussian blur In image processing, a Gaussian blur (also known as Gaussian smoothing) is the result of blurring an image by a Gaussian function (named after mathematician and scientist Carl Friedrich Gauss). It is a widely used effect in graphics software, ...
*
Gaussian filter In electronics and signal processing, mainly in digital signal processing, a Gaussian filter is a filter (signal processing), filter whose impulse response is a Gaussian function (or an approximation to it, since a true Gaussian response would h ...
* Husimi Q representation * Heat equation#Fundamental solutions


Notes


References

{{DEFAULTSORT:Weierstrass Transform Integral transforms Mathematical physics