 Finite-rank projection kernel

An orthonormal set ${\varphi }_0,\dots ,{\varphi }_{m-1}$ in an $L^2$ space defines the integral kernel of the orthogonal projection onto its span by this sum. It satisfies $\int K(x,y)K(y,z)d\mu (y)=K(x,z)$ and $\int K(x,x)d\mu (x)=m$. Its evaluation matrices are positive semidefinite Gram matrices.