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Concept
Integral transform
Summary
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.
An integral transform is any transform of the following form:
The input of this transform is a function , and the output is another function . 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 of two variables, the kernel function, integral kernel or nucleus of the transform.
Some kernels have an associated inverse kernel which (roughly speaking) yields an inverse transform:
A symmetric kernel is one that is unchanged when the two variables are permuted; it is a kernel function such that . In the theory of integral equations, symmetric kernels correspond to self-adjoint operators.
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" or stochastic discount factor, or the smoothing of data recovered from robust statistics; see kernel (statistics).
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.
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