reproducingkernel

A reproducing kernel, in the context of functional analysis, is a function K: X × X → F associated with a reproducing kernel Hilbert space H of functions on X. It has the reproducing property: for each x in X there exists a function Kx(·) in H such that f(x) = ⟨f, Kx⟩H for every f in H. The kernel is symmetric and positive definite, and K(x, y) = ⟨Ky, Kx⟩H.

This property implies that evaluation at a point is a continuous linear functional on H, and the set spanned by {Kx : x ∈ X} is dense in H, so H is the closure of that span. For any finite subset {x1, ..., xn} of X, the Gram matrix [K(xi, xj)] is positive semidefinite. Conversely, every positive definite kernel on X determines an RKHS via the Moore–Aronszajn construction.

Common examples include the linear kernel K(x, y) = x·y, the polynomial kernel (x·y + c)^d, and the Gaussian (or radial basis function) kernel K(x, y) = exp(-||x − y||^2 / (2σ^2)). These kernels induce corresponding RKHSs of functions on X.

In practice, reproducing kernels underpin kernel methods in machine learning, such as support vector machines, kernel ridge regression, and kernel PCA, enabling nonlinear learning through inner products in high-dimensional feature spaces. In statistics, kernels serve as covariance functions of Gaussian processes; the associated RKHS relates to the Cameron–Martin space, connecting probabilistic and functional-analytic perspectives. The concept originated with Aronszajn’s theory of reproducing kernel Hilbert spaces and remains central to the kernel trick, which computes inner products in feature spaces without explicit mappings.