Nyström quadrature operators and their reconstructed equations

Let $X:=C([0,1],\mathbb{R})$ be the real continuous functions on $[0,1]$ (The space $C(K,\mathbb{R})$ of continuous real-valued functions on a nonempty compact metric space) with the supremum norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}:={\sup }_{0\le x\le 1}\mathrm{\mid }u(x)\mathrm{\mid }$. The interval $[0,1]$ is a nonempty compact metric space by Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line. Pointwise addition and real scalar multiplication preserve continuity. The supremum is finite by $C(K,\mathbb{R})$ is complete in the supremum metric for every nonempty compact metric space $K$; the pointwise absolute-value inequalities give $\mathrm{\parallel }u+v{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}+\mathrm{\parallel }v{\mathrm{\parallel }}_{\mathrm{\infty }}$ and $\mathrm{\parallel }cu{\mathrm{\parallel }}_{\mathrm{\infty }}=\mathrm{\mid }c\mathrm{\mid }\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}$, and $\mathrm{\parallel }u{\mathrm{\parallel }}_{\mathrm{\infty }}=0$ holds exactly when $u=0$. Thus this is a real norm (A norm on a real vector space, the induced metric, and the dictionary with the metric axioms). Its induced metric is the complete supremum metric of $C(K,\mathbb{R})$ is complete in the supremum metric for every nonempty compact metric space $K$, so $X$ is Banach (Banach space). Fix a continuous real kernel $k:[0,1{]}^2\to \mathbb{R}$. The integral ${\int }_0^1$ below is the ordinary Riemann integral, equivalently the Riemann–Stieltjes integral with integrator $y\mapsto y$ (Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral). For $u\in X$, define $(Ku)(x):={\int }_0^{1}k(x,y)u(y)dy,0\le x\le 1.$ The integrand is continuous in $y$, so this integral exists by A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion.

A finite quadrature rule on $[0,1]$ consists of an integer $m\ge 1$, nodes $y_1,\dots ,y_m\in [0,1]$, and real weights $w_1,\dots ,w_m$. It acts on a function $g$ defined at the nodes by $Q(g):={\sum }_{j=1}^m{w}_{j}g(y_j)\mathrm{.}$ Its Nyström operator is the finite-sum rule $(K_{Q}u)(x):=Q(k(x,\cdot )u(\cdot ))={\sum }_{j=1}^m{w}_{j}k(x,y_j)u(y_j)\mathrm{.}$ In particular, for the composite midpoint rule with $N\ge 1$ panels, $h:=1\mathrm{/}N$, $y_j:=(j+\frac{1}{2})h$ for $0\le j<N$, and $w_j=h$, the operator is denoted $K_N$. The nodes and weights are those of Composite midpoint rule on a uniform grid.

For a right-hand side $f\in X$, the Nyström equation is $u_Q-K_Q{u}_Q=f\mathrm{.}$ This definition asserts neither consistency nor convergence of $K_Q$ to $K$.