nLab
integral transform

Idea

An integral transform on functions is a linear map between functions on spaces $X$, $Y$ encoded by a function $K$ on the product space $X\times Y$ and given by a formula of the type

$$(\mathrm{function}f\mathrm{on}X)\mapsto (y\mapsto {\int }_{X}f(x)K(x,y)),$$(function f on X) \mapsto \left(y \mapsto \int_{X} f(x) K(x,y) \right) \,,

where on the right we have some kind of integration over $X$.

Here $K$ is called the integral kernel of the transformation.

Typically the definition of an integral transform on functions involves some delicate technical issues concerning the precise nature of the function space, the measure with respect to which the integral is defined, etc.

On the other hand, one may understand the general form of an integral transform as the decategorification of a very natural general abstract construction in higher category theory: that of integral transforms on sheaves given by spans of base change geometric morphisms.

Special cases of such categorified integral transforms are discussed at

• Fourier-Mukai transform, topological T-duality

• groupoidification

• geometric infinity-function theory.

Examples

The simplest example is matrix multiplication, which corresponds to the case where $X$ and $Y$ are discrete spaces.

Standard examples involving genuine functional analysis are for instance the Fourier transform or the Laplace transform?.

Also the path integral in quantum mechanics and quantum field theory is supposed to be a class of examples of an integral kernel.