A symmetric function $k:\mathcal X\times\mathcal X\to\mathbb R$ is a <positive-semidefinite kernel> when, for every $m$, every $x_1,\ldots,x_m\in\mathcal X$, and every $c\in\mathbb R^m$,
$$
\sum_{i,j=1}^mc_ic_jk(x_i,x_j)\geq0.
$$
Equivalently, every associated <kernel matrix> is a <positive semidefinite matrix>.

A <Reproducing-kernel Hilbert space> $\mathcal H$ on $\mathcal X$ is a <Hilbert space> of real-valued functions such that every point-evaluation map $f\mapsto f(x)$ is continuous. The <Riesz representation theorem> then gives a function $k(x,\cdot)\in\mathcal H$ satisfying the <reproducing property>
$$
f(x)=\langle f,k(x,\cdot)\rangle_{\mathcal H}.
$$
Its reproducing kernel is $k(x,y)=\langle k(x,\cdot),k(y,\cdot)\rangle_{\mathcal H}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 205 / 1 / a / Solution

The reproducing property is $f(x)=\langle f,k(x,\cdot)\rangle_{\mathcal H}$; it turns point evaluation into an inner product with a kernel section.


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 Incoming links: Reproducing property