Kernel (mathematics)

In mathematics, the word kernel has several meanings. Kernel may mean a subset associated with a mapping:

• The kernel of a mapping is the set of elements that map to the zero element (such as zero or zero vector), as in kernel of a linear operator and kernel of a matrix. In this context, kernel is often called nullspace.
• More generally, the kernel in algebra is the set of elements that map to the neutral element. Here, the mapping is assumed to be a homomorphism, that is, it preserves algebraic operations, and, in particular, maps neutral element to neutral element. The kernel is then the set of all elements that the mapping cannot distinguish from the neutral element.
• The kernel in category theory is a generalization of this concept to morphisms rather than mappings between sets.
• In set theory, the kernel of a function is the set of all pairs of elements that the function cannot distinguish, that is, they map to the same value. This is a generalization of the kernel concept above to the case when there is no neutral element.
• In set theory, the difference kernel or binary equalizer is the set of all elements where the values of two functions coincide.

Kernel may also mean a function of two variables, which is used to define a mapping:

• In integral calculus, the kernel (also called integral kernel or kernel function) is a function of two variables that defines an integral transform, such as the function k in

$(T f)(x) = \int_X k(x, x') f(x') \, dx'.$

• In partial differential equations, when the solution of the equation for the right-hand side f can be written as Tf above, the kernel becomes the Green's function. The heat kernel is the Green's function of the heat equation.
• In the case when the integral kernel depends only on the difference between its arguments, it becomes a convolution kernel, as in

$(T f)(x) = \int_X \phi(x - x') f(x') \, dx'.$

• In statistics, a kernel is a weighting function used in kernel density estimation to estimate the probability density function of a random variable.
• In probability theory and statistics, a stochastic kernel is the transition function of a stochastic process. In a discrete time process with continuous probability distributions, it is the same thing as the kernel of the integral operator that advances the probability density function.
• Kernel trick is a technique to write a nonlinear operator as a linear one in a space of higher dimension.
• In operator theory, a positive definite kernel is a generalization of a positive-definite matrix.
• The kernel in a reproducing kernel Hilbert space.