Convergence of approximations for second-kind equations

Remarks

The solution $\varphi$ exists because, for compact $K$, injectivity of $I - K$ implies that $I - K$ is invertible with bounded inverse (Riesz theory, the Fredholm alternative). Since $P_n \psi \to \psi$ for each $\psi$, the norms $\norm{P_n}$ are bounded by the uniform boundedness principle, so the convergence is uniform on compact sets. The set $\set{K\psi : \norm{\psi} \le 1}$ is relatively compact, hence $\norm{P_n K - K} \to 0$. A perturbation argument then shows that $I - P_n K$ is invertible for large $n$, with $\norm{(I - P_n K)^{-1}} \le M$. Finally $(I - P_n K)(\varphi_n - \varphi) = P_n \varphi - \varphi$, which gives the error bound; and if $\varphi_n - P_nK\varphi_n = P_n f$ then $\varphi_n = P_n(f + K\varphi_n) \in X_n$.

Example: for a continuous kernel on $[a, b]$, collocation with piecewise linear interpolation on grids with mesh tending to $0$ satisfies the hypothesis in $C[a, b]$, since $P_n \psi \to \psi$ uniformly for continuous $\psi$. The error is then of the order of the interpolation error of the exact solution, for instance $O(h^2)$ for a twice continuously differentiable $\varphi$ and mesh size $h$.

The hypothesis of pointwise convergence $P_n \psi \to \psi$ is needed. For collocation with polynomial interpolation at equidistant points in $C[a, b]$ it fails, since the norms of the interpolation operators grow without bound, and the theorem gives nothing.

Going deeper

For a student

The key step is that a compact operator turns pointwise convergence of $P_n$ into norm convergence of $P_n K$. This is why second-kind equations with compact $K$ are so well suited to numerical methods, in contrast to first-kind equations, which are ill-posed. In a Hilbert space with orthogonal projections, this gives the convergence of the Galerkin method for second-kind equations. For the Nyström method, $K$ is replaced by quadrature operators $K_n$ that do not converge in norm, and the same conclusion needs collectively compact operator approximation.