Integrotransformation

Integrotransformation is a broad term used to describe a transformation that maps a function to another function through an integral operator. In its standard formulation, the transformed function g is obtained from an input function f by g(x) = ∫_a^b K(x,t) f(t) dt, where K(x,t) is a kernel that encodes how values of f contribute to the output at x. Some formulations allow the kernel to depend on derivatives of f, producing integro-differential transforms, for example when K involves terms with d^n f/dt^n inside the integral.

These transformations are linear when the kernel is fixed and the domain of integration is fixed. They are studied within functional analysis and operator theory, with attention to questions of boundedness, range, and invertibility. An inverse transform, when it exists, often takes the form f(t) = ∫_a^b L(t,x) g(x) dx, where L is an inverse or adjoint kernel. Existence and construction of an inverse are tied to properties of the kernel, such as completeness, symmetry, and the solvability of associated Fredholm or Volterra integral equations.

Integrotransformations relate to and encompass many classical integral transforms. If the kernel depends only on x − t, the operator resembles a convolution and may link to transforms used in signal processing. More familiar examples of integral transforms, such as the Fourier, Laplace, and Hankel transforms, can be viewed as specific kernels within the same general framework, each with its own inversion formulas and application domain.

Applications include solving integral and integro-differential equations, boundary-value problems, problems with memory effects in physics and engineering, and various problems in probability and statistics.