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. The transformed function can generally be mapped back to the original function space using the inverse transform.
General form
An integral transform is any transform T of the following form: 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 variables, the kernel function, integral kernel or nucleus of the transform. Some kernels have an associated inverse kernelK−1 which yields an inverse transform: A symmetric kernel is one that is unchanged when the two variables are permuted; it is a kernel function K such that K = K.
Motivation for use
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" into another domain. Manipulating and solving the equation in the target domain can be 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. 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.
History
The precursor of the transforms were the Fourier series to express functions in finite intervals. Later the Fourier transform was developed to remove the requirement of finite intervals. Using the Fourier series, just about any practical function of time can be represented as a sum of sines and cosines, each suitably scaled, shifted and "squeezed" or "stretched". The sines and cosines in the Fourier series are an example of an orthonormal basis.
Usage example
As an example of an application of integral transforms, consider the Laplace transform. 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. The equation cast in terms of complex frequency is readily solved in the complex frequency domain, leading to a "solution" formulated in the frequency domain. Employing the inverse 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 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 equations 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. Another usage example is the kernel in path integral: This states that the total amplitude to arrive at is the sum, or the integral, over all possible values of of the total amplitude to arrive at the point multiplied by the amplitude to go from x' to x that is,. It is often referred to as the [propagator of a given system. This kernel is the kernel of integral transform. However, for each quantum system, there is a different kernel.
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. 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, integration kernels are then biperiodic functions; convolution by functions on the circle yields circular convolution.
If one uses functions on the cyclic group of order n, one obtains n × n matrices as integration kernels; convolution corresponds to circulant matrices.
