Collocation tableau from distinct quadrature nodes

Let $M,d\ge 1$, let $0\le c_1<\cdots <c_M\le 1$ be distinct collocation nodes, and let ${\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_M$ be their cardinal polynomials from Cardinal-polynomial interpolation. Define $Q_{ij}:={\int }_0^{c_i}{\mathrm{\ell }}_j(s)ds,b_j:={\int }_0^1{\mathrm{\ell }}_j(s)ds\mathrm{.}$ The matrix $Q\in {\mathbb{R}}^{M\times M}$ and vector $b\in {\mathbb{R}}^M$ are the collocation tableau coefficients. For an IVP $y^{\mathrm{\prime }}=f(t,y)$ with $f:D\to {\mathbb{R}}^d$ on the open time-state domain $D$ of First-order systems, initial value problems, and solutions on intervals, fix $(t,u)\in D$ and a step $h>0$. A stage vector $U\in ({\mathbb{R}}^d{)}^M$ is admissible only when $(t+c_{j}h,U_j)\in D$ for every $j$. On this admissible set write $F_j(U):=f(t+c_{j}h,U_j),QF(U):={({\sum }_{j=1}^M{Q}_{ij}{F}_j(U))}_{i=1}^M\mathrm{.}$ All displayed integrals use the oriented Riemann convention of The integral with oriented limits: ${\int }_a^{a}f:=0$ and ${\int }_b^{a}f:=-{\int }_a^{b}f$. In particular $Q_{ij}=0$ when $c_i=0$, and the integral in $u_h$ vanishes at $\tau =0$. The collocation stages and endpoint are the implicit Runge–Kutta equations of Implicit Runge–Kutta stage equations and update with $A=Q$, nodes $c$, and weights $b$: $U=\mathbf{1}u+hQF(U),{\mathrm{\Psi }}_h(t,u)=u+h{\sum }_{j=1}^M{b}_j{F}_j(U)\mathrm{.}$ The associated collocation polynomial on normalized time $\tau \in [0,1]$ is $u_h(\tau ):=u+h{\int }_0^{\tau }{\sum }_{j=1}^M{\mathrm{\ell }}_j(s)F_j(U)ds\mathrm{.}$ Thus $u_h(c_i)=U_i$, $u_h^{\mathrm{\prime }}(c_i)=h{F}_i(U)$, and $u_h(1)={\mathrm{\Psi }}_h(t,u)$ whenever $U$ is an admissible solution. This definition alone asserts no branch existence or uniqueness.

For interpolation estimates define the Lebesgue constant of this fixed node set by ${\mathrm{\Lambda }}_M:={\max }_{0\le s\le 1}{\sum }_{j=1}^M\mathrm{\mid }{\mathrm{\ell }}_j(s)\mathrm{\mid }\mathrm{.}$ For a real matrix $A$, write $\mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{\infty }}:={\max }_i{\sum }_j\mathrm{\mid }{a}_{ij}\mathrm{\mid }$. These constants depend on the nodes and on $M$; no uniform-in-$M$ bound is part of the definition.