Integral transform

Inmathematics,anintegral transformis a type oftransformthat maps afunctionfrom its originalfunction spaceinto another function space viaintegration,where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using theinverse transform.

General form

An integral transform is anytransform $T$ of the following form:

(Tf)(u)=\int _{t_{1}}^{t_{2}}f(t)\,K(t,u)\,dt

The input of this transform is afunction $f$ ,and the output is another function $Tf$ .An integral transform is a particular kind of mathematicaloperator.

There are numerous useful integral transforms. Each is specified by a choice of the function $K$ of twovariables,that is called thekernelornucleusof the transform.

Some kernels have an associatedinverse kernel $K^{-1}(u,t)$ which (roughly speaking) yields an inverse transform:

f(t)=\int _{u_{1}}^{u_{2}}(Tf)(u)\,K^{-1}(u,t)\,du

Asymmetric kernelis one that is unchanged when the two variables are permuted; it is a kernel function $K$ such that $K(t,u)=K(u,t)$ .In the theory of integral equations, symmetric kernels correspond toself-adjoint operators.^[1]

Motivation

There are many classes of problems that are difficult to solve—or at least quite unwieldy algebraically—in their original representations. An integral transform "maps" an equation from its original "domain" into another domain, in which manipulating and solving the equation may be much easier than in the original domain. The solution can then be mapped back to the original domain with the inverse of the integral transform.

There are many applications of probability that rely on integral transforms, such as "pricing kernel" orstochastic discount factor,or the smoothing of data recovered from robust statistics; seekernel (statistics).

History

The precursor of the transforms were theFourier seriesto express functions in finite intervals. Later theFourier transformwas developed to remove the requirement of finite intervals.

Using the Fourier series, just about any practical function of time (thevoltageacross the terminals of anelectronic devicefor example) can be represented as a sum ofsinesandcosines,each suitably scaled (multiplied by a constant factor), shifted (advanced or retarded in time) and "squeezed" or "stretched" (increasing or decreasing the frequency). The sines and cosines in the Fourier series are an example of anorthonormal basis.

Usage example

As an example of an application of integral transforms, consider theLaplace transform.This is a technique that mapsdifferentialorintegro-differential equationsin the"time" domaininto polynomial equations in what is termed the"complex frequency" domain.(Complex frequency is similar to actual, physical frequency but rather more general. Specifically, the imaginary componentωof the complex frequencys= −σ+iωcorresponds to the usual concept of frequency,viz.,the rate at which a sinusoid cycles, whereas the real componentσof the complex frequency corresponds to the degree of "damping", i.e. an exponential decrease of the amplitude.) The equation cast in terms of complex frequency is readily solved in the complex frequency domain (roots of the polynomial equations in the complex frequency domain correspond toeigenvaluesin the time domain), leading to a "solution" formulated in the frequency domain. Employing theinverse transform,i.e.,the inverse procedure of the original Laplace transform, one obtains a time-domain solution. In this example, polynomials in the complex frequency domain (typically occurring in the denominator) correspond topower seriesin the time domain, while axial shifts in the complex frequency domain correspond to damping by decaying exponentials in the time domain.

The Laplace transform finds wide application in physics and particularly in electrical engineering, where thecharacteristic equationsthat describe the behavior of an electric circuit in the complex frequency domain correspond to linear combinations of exponentially scaled and time-shifteddamped sinusoidsin the time domain. Other integral transforms find special applicability within other scientific and mathematical disciplines.

Another usage example is the kernel in thepath integral:

\psi (x,t)=\int _{-\infty }^{\infty }\psi (x',t')K(x,t;x',t')dx'.

This states that the total amplitude $\psi (x,t)$ to arrive at $(x,t)$ is the sum (the integral) over all possible values $x'$ of the total amplitude $\psi (x',t')$ to arrive at the point $(x',t')$ multiplied by the amplitude to go from $x'$ to $x$ [i.e. $K(x,t;x',t')$ ].^[2]It is often referred to as thepropagatorfor a given system. This (physics) kernel is the kernel of the integral transform. However, for each quantum system, there is a different kernel.^[3]

Table of transforms

Table of integral transforms
Transform	Symbol	K	f(t)	t₁	t₂	K⁻¹	u₁	u₂
Abel transform	F, f	${\frac {2t}{\sqrt {t^{2}-u^{2}}}}$		$u$	$\infty$	${\frac {-1}{\pi {\sqrt {u^{2}\!-\!t^{2}}}}}{\frac {d}{du}}$ ^[4]	t	$\infty$
Associated Legendre transform	${\mathcal {J}}_{n,m}$	$(1-x^{2})^{-m/2}P_{n}^{m}(x)$		$-1$	$1$		$0$	$\infty$
Fourier transform	${\mathcal {F}}$	$e^{-2\pi iut}$	$L_{1}$	$-\infty$	$\infty$	$e^{2\pi iut}$	$-\infty$	$\infty$
Fourier sine transform	${\mathcal {F}}_{s}$	${\sqrt {\frac {2}{\pi }}}\sin(ut)$	on $[0,\infty )$ ,real-valued	$0$	$\infty$	${\sqrt {\frac {2}{\pi }}}\sin(ut)$	$0$	$\infty$
Fourier cosine transform	${\mathcal {F}}_{c}$	${\sqrt {\frac {2}{\pi }}}\cos(ut)$	on $[0,\infty )$ ,real-valued	$0$	$\infty$	${\sqrt {\frac {2}{\pi }}}\cos(ut)$	$0$	$\infty$
Hankel transform		$t\,J_{\nu }(ut)$		$0$	$\infty$	$u\,J_{\nu }(ut)$	$0$	$\infty$
Hartley transform	${\mathcal {H}}$	${\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}$		$-\infty$	$\infty$	${\frac {\cos(ut)+\sin(ut)}{\sqrt {2\pi }}}$	$-\infty$	$\infty$
Hermite transform	$H$	$e^{-x^{2}}H_{n}(x)$		$-\infty$	$\infty$		$0$	$\infty$
Hilbert transform	${\mathcal {H}}il$	${\frac {1}{\pi }}{\frac {1}{u-t}}$		$-\infty$	$\infty$	${\frac {1}{\pi }}{\frac {1}{u-t}}$	$-\infty$	$\infty$
Jacobi transform	$J$	$(1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha,\beta }(x)$		$-1$	$1$		$0$	$\infty$
Laguerre transform	$L$	$e^{-x}\ x^{\alpha }\ L_{n}^{\alpha }(x)$		$0$	$\infty$		$0$	$\infty$
Laplace transform	${\mathcal {L}}$	$e^{-ut}$		$0$	$\infty$	${\frac {e^{ut}}{2\pi i}}$	$c\!-\!i\infty$	$c\!+\!i\infty$
Legendre transform	${\mathcal {J}}$	$P_{n}(x)\,$		$-1$	$1$		$0$	$\infty$
Mellin transform	${\mathcal {M}}$	$t^{u-1}$		$0$	$\infty$	${\frac {t^{-u}}{2\pi i}}\,$ ^[5]	$c\!-\!i\infty$	$c\!+\!i\infty$
Two-sided Laplace transform	${\mathcal {B}}$	$e^{-ut}$		$-\infty$	$\infty$	${\frac {e^{ut}}{2\pi i}}$	$c\!-\!i\infty$	$c\!+\!i\infty$
Poisson kernel		${\frac {1-r^{2}}{1-2r\cos \theta +r^{2}}}$		$0$	$2\pi$
Radon transform	Rƒ			$-\infty$	$\infty$
Weierstrass transform	${\mathcal {W}}$	${\frac {e^{-{\frac {(u-t)^{2}}{4}}}}{\sqrt {4\pi }}}\,$		$-\infty$	$\infty$	${\frac {e^{\frac {(u-t)^{2}}{4}}}{i{\sqrt {4\pi }}}}$	$c\!-\!i\infty$	$c\!+\!i\infty$
X-ray transform	Xƒ			$-\infty$	$\infty$

In the limits of integration for the inverse transform,cis a constant which depends on the nature of the transform function. For example, for the one and two-sided Laplace transform,cmust be greater than the largest real part of the zeroes of the transform function.

Note that there are alternative notations and conventions for the Fourier transform.

Different domains

Here integral transforms are defined for functions on the real numbers, but they can be defined more generally for functions on a group.

If instead one uses functions on the circle (periodic functions), integration kernels are then biperiodic functions; convolution by functions on the circle yieldscircular convolution.
If one uses functions on thecyclic groupof ordern( $C n$ or $Z / n Z$ ), one obtainsn×nmatrices as integration kernels; convolution corresponds tocirculant matrices.

General theory

Although the properties of integral transforms vary widely, they have some properties in common. For example, every integral transform is alinear operator,since the integral is a linear operator, and in fact if the kernel is allowed to be ageneralized functionthen all linear operators are integral transforms (a properly formulated version of this statement is theSchwartz kernel theorem).

The general theory of suchintegral equationsis known asFredholm theory.In this theory, the kernel is understood to be acompact operatoracting on aBanach spaceof functions. Depending on the situation, the kernel is then variously referred to as theFredholm operator,thenuclear operatoror theFredholm kernel.

References

^Chapter 8.2, Methods of Theoretical Physics Vol. I (Morse & Feshbach)
^Eq 3.42 in Feynman and Hibbs, Quantum Mechanics and Path Integrals, emended edition:
^Mathematically, what is the kernel in path integral?
^Assuming the Abel transform is not discontinuous at $u$ .
^Some conditions apply, seeMellin inversion theoremfor details.