An integral transform is a particular kind of mathematical operator.
In the theory of integral equations, symmetric kernels correspond to self-adjoint operators.
[1] There are many classes of problems that are difficult to solve—or at least quite unwieldy algebraically—in their original representations.
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.
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).
The sines and cosines in the Fourier series are an example of an orthonormal basis.
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.
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.)
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.
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.
Other integral transforms find special applicability within other scientific and mathematical disciplines.
Another usage example is the kernel in the path integral: This states that the total amplitude
Note that there are alternative notations and conventions for the Fourier transform.
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).
In this theory, the kernel is understood to be a compact operator acting on a Banach space of functions.