Integral transform

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For other uses, see Transformation (mathematics). Integral transform_sentence_0

In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. Integral transform_sentence_1

The transformed function can generally be mapped back to the original function space using the inverse transform. Integral transform_sentence_2

General form Integral transform_section_0

An integral transform is any transform T of the following form: Integral transform_sentence_3

The input of this transform is a function f, and the output is another function Tf. Integral transform_sentence_4

An integral transform is a particular kind of mathematical operator. Integral transform_sentence_5

There are numerous useful integral transforms. Integral transform_sentence_6

Each is specified by a choice of the function K of two variables, the kernel function, integral kernel or nucleus of the transform. Integral transform_sentence_7

Some kernels have an associated inverse kernel K(u, t) which (roughly speaking) yields an inverse transform: Integral transform_sentence_8

A symmetric kernel is one that is unchanged when the two variables are permuted; it is a kernel function K such that K(t, u) = K(u, t). Integral transform_sentence_9

Motivation for use Integral transform_section_1

Mathematical notation aside, the motivation behind integral transforms is easy to understand. Integral transform_sentence_10

There are many classes of problems that are difficult to solve—or at least quite unwieldy algebraically—in their original representations. Integral transform_sentence_11

An integral transform "maps" an equation from its original "domain" into another domain. Integral transform_sentence_12

Manipulating and solving the equation in the target domain can be much easier than manipulation and solution in the original domain. Integral transform_sentence_13

The solution is then mapped back to the original domain with the inverse of the integral transform. Integral transform_sentence_14

There are many applications of probability that rely on integral transforms, such as "pricing kernel" or stochastic discount factor, or the smoothing of data recovered from robust statistics; see kernel (statistics). Integral transform_sentence_15

History Integral transform_section_2

The precursor of the transforms were the Fourier series to express functions in finite intervals. Integral transform_sentence_16

Later the Fourier transform was developed to remove the requirement of finite intervals. Integral transform_sentence_17

Using the Fourier series, just about any practical function of time (the voltage across the terminals of an electronic device for example) can be represented as a sum of sines and cosines, each suitably scaled (multiplied by a constant factor), shifted (advanced or retarded in time) and "squeezed" or "stretched" (increasing or decreasing the frequency). Integral transform_sentence_18

The sines and cosines in the Fourier series are an example of an orthonormal basis. Integral transform_sentence_19

Usage example Integral transform_section_3

As an example of an application of integral transforms, consider the Laplace transform. Integral transform_sentence_20

This is a technique that maps differential or integro-differential equations in the "time" domain into polynomial equations in what is termed the "complex frequency" domain. Integral transform_sentence_21

(Complex frequency is similar to actual, physical frequency but rather more general. Integral transform_sentence_22

Specifically, the imaginary component ω of the complex frequency s = -σ + iω corresponds to the usual concept of frequency, viz., the rate at which a sinusoid cycles, whereas the real component σ of the complex frequency corresponds to the degree of "damping", i.e. an exponential decrease of the amplitude.) Integral transform_sentence_23

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 to eigenvalues in the time domain), leading to a "solution" formulated in the frequency domain. Integral transform_sentence_24

Employing the inverse transform, i.e., the inverse procedure of the original Laplace transform, one obtains a time-domain solution. Integral transform_sentence_25

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. Integral transform_sentence_26

The Laplace transform finds wide application in physics and particularly in electrical engineering, where the characteristic equations that describe the behavior of an electric circuit in the complex frequency domain correspond to linear combinations of exponentially scaled and time-shifted damped sinusoids in the time domain. Integral transform_sentence_27

Other integral transforms find special applicability within other scientific and mathematical disciplines. Integral transform_sentence_28

Another usage example is the kernel in path integral: Integral transform_sentence_29

Table of transforms Integral transform_section_4

In the limits of integration for the inverse transform, c is a constant which depends on the nature of the transform function. Integral transform_sentence_30

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. Integral transform_sentence_31

Note that there are alternative notations and conventions for the Fourier transform. Integral transform_sentence_32

Different domains Integral transform_section_5

Here integral transforms are defined for functions on the real numbers, but they can be defined more generally for functions on a group. Integral transform_sentence_33

General theory Integral transform_section_6

Although the properties of integral transforms vary widely, they have some properties in common. Integral transform_sentence_34

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 a generalized function then all linear operators are integral transforms (a properly formulated version of this statement is the Schwartz kernel theorem). Integral transform_sentence_35

The general theory of such integral equations is known as Fredholm theory. Integral transform_sentence_36

In this theory, the kernel is understood to be a compact operator acting on a Banach space of functions. Integral transform_sentence_37

Depending on the situation, the kernel is then variously referred to as the Fredholm operator, the nuclear operator or the Fredholm kernel. Integral transform_sentence_38

See also Integral transform_section_7

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: transform.