Integral transform

Mapping involving integration between function spaces In mathematics, an integral transform is a type of transformation that 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 {\displaystyle T} of the following form: ( T f ) ( u ) = ∫ t 1 t 2 f ( t ) K ( t , u ) d t {\displaystyle (Tf)(u)=\int _{t_{1}}^{t_{2}}f(t)\,K(t,u)\,dt} The input of this transform is a function f {\displaystyle f} , and the output is another function T f {\displaystyle 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 {\displaystyle K} of two variables, that is called the kernel or nucleus of the transform. Some kernels have an associated inverse kernel K − 1 ( u , t ) {\displaystyle K^{-1}(u,t)} which (roughly speaking) yields an inverse t