- Integral transform
In

mathematics , an**integral transform**is any transform "T" of the following form::$(Tf)(u)\; =\; int\_\{t\_1\}^\{t\_2\}\; K(t,\; u),\; f(t),\; dt.$

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

operator .There are numerous useful integral transforms. Each is specified by a choice of the function "K" of two

variable s, the**kernel function**or**nucleus**of the transform.Some kernels have an associated "inverse kernel" $K^\{-1\}(\; u,t\; )$ which (roughly speaking) yields an inverse transform:

:$f(t)\; =\; int\_\{u\_1\}^\{u\_2\}\; K^\{-1\}(\; u,t\; ),\; (Tf)(u),\; du.$

A "symmetric kernel" is one that is unchanged when the two variables are permuted.

**Motivation**Mathematical notation aside, the motivation behind integral transforms is easy to understand. 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" ("e.g.", functions where time is the independent variable are said to be in the

time domain ) into another domain. Manipulating and solving the equation in the target domain is, ideally, much easier than manipulation and solution in the original domain. The solution is then mapped back to the original domain with the inverse of the integral transform.Integral transforms work because they are based upon the concept of "spectral factorization" over orthonormal bases. What this means is that many important arbitrarily complicated functions can be represented as sums of much simpler functions.

**History**The precursor of the transforms were the

Fourier series to express functions in finite intervals. Later theFourier transform was developed to remove the requirement of finite intervals.Using the Fourier series, just about any practical function of time (the

voltage across the terminals of anelectronic device for example) can be represented as a sum ofsine s andcosine s, each suitably scaled (multiplied by a constant factor) and shifted (advanced or retarded in time). The sines and cosines in the Fourier series are an example of an orthonormal basis.**Importance of orthogonality**The individual basis functions have to be

orthogonal . That is, the product of two dissimilar basis functions—integrated over their domain—must be zero. An integral transform, in actuality, just changes the representation of a function from one orthogonal basis to another. Each point in the representation of the transformed function in the target domain corresponds to the contribution of a given orthogonal basis function to the expansion. The process of expanding a function from its "standard" representation to a sum of a number of orthonormal basis functions, suitably scaled and shifted, is termed "spectral factorization ." This is similar in concept to the description of a point in space in terms of three discrete components, namely, its "x", "y", and "z" coordinates. Each axis correlates only to itself and nothing to the other orthogonal axes. Note the terminological consistency: the determination of the amount by which an individual orthonormal basis function must be scaled in the spectral factorization of a function, "F", is termed the "projection" of "F" onto that basis function.The normal Cartesian graph "per se" of a function can be thought of as an orthonormal expansion. Indeed, each point just reflects the contribution of a given orthonormal basis function to the sum. Intuitively, the point (3,5) on the graph means that the orthonormal basis function δ(x-3), where "δ" is the

Dirac delta function , is scaled up by a factor of five to contribute to the sum in this form. In this way, the graph of a continuous real-valued function in the plane corresponds to an "infinite" set of basis functions; if the number of basis functions were finite, the curve would consist of a discrete set of points rather than a continuous contour.**Usage example**As an example of an application of integral transforms, consider the

Laplace transform . This is a technique that maps differential orintegro-differential equation s in the "time" domain into 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 component "ω" of the complex frequency "s = -σ + iω" corresponds to the usual concept of frequency, "viz.", the speed at which a sinusoid cycles, whereas the real component "σ" of the complex frequency corresponds to the degree of "damping". ) 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 toeigenvalues in 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 to power series in 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 the

characteristic equation s that describe the behavior of an electric circuit in the complex frequency domain correspond to linear combinations of exponentially damped, scaled, and time-shifted sinusoids in the time domain. Other integral transforms find special applicability within other scientific and mathematical disciplines.**Table of transforms**In the limits of integration for the inverse transform, "c" is a constant which depends on the nature of the transform function. For example, for the one and two-sided Laplace transform, "c" must be greater than the largest real part of the zeroes of the transform function.

**General theory**Although the properties of integral transforms vary widely, they have some properties in common. For example, every integral transform is a

linear operator , since the integral is a linear operator, and in fact if the kernel is allowed to be ageneralized function then all linear operators are integral transforms (a properly formulated version of this statement is theSchwartz kernel theorem ).The general theory of such

integral equation s is known asFredholm theory . In this theory, the kernel is understood to be acompact operator acting on aBanach space of functions. Depending on the situation, the kernel is then variously referred to as theFredholm operator , thenuclear operator or theFredholm kernel .**See also***

Convolution kernel

*List of transforms

*List of operators

*List of Fourier-related transforms

*Kernel trick

*Kernel methods

*Nachbin's theorem **References*** A. D. Polyanin and A. V. Manzhirov, "Handbook of Integral Equations", CRC Press, Boca Raton, 1998. ISBN 0-8493-2876-4

* [*http://eqworld.ipmnet.ru/en/auxiliary/aux-inttrans.htm Tables of Integral Transforms*] at EqWorld: The World of Mathematical Equations.

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