Standard midpoint Nyström operators need not converge in operator norm on C

Let $m_j:=(j+\frac{1}{2})h$. Define $u_{\rho }$ to equal $-1+2\mathrm{\mid }y-m_j\mathrm{\mid }\mathrm{/}\rho$ on each interval $\mathrm{\mid }y-m_j\mathrm{\mid }\le \rho$ and $1$ elsewhere. The windows are disjoint and lie inside $[0,1]$, so $u_{\rho }$ is continuous and $\mathrm{\parallel }{u}_{\rho }{\mathrm{\parallel }}_{\mathrm{\infty }}=1$. Its integral over each window is zero, hence ${\int }_0^1{u}_{\rho }(y)dy=1-2N\rho$.

Every midpoint value is $u_{\rho }(m_j)=-1$, so $K_N{u}_{\rho }=-1$ by [F1] and [F2], whereas $K{u}_{\rho }=1-2N\rho$. Therefore $\mathrm{\parallel }(K-K_N)u_{\rho }{\mathrm{\parallel }}_{\mathrm{\infty }}=2-2N\rho$.

The operators $K$ and $K_N$ both have norm at most $1$, because their defining integrals or positive quadrature weights have total mass $1$. Thus $\mathrm{\parallel }K-K_N\mathrm{\parallel }\le 2$. Letting $\rho \downarrow 0$ in step 2.1 gives $\mathrm{\parallel }K-K_N\mathrm{\parallel }\ge 2$, so $\mathrm{\parallel }K-K_N\mathrm{\parallel }=2$ for every $N$. This disproves the claimed operator-norm convergence.