Nested one-dimensional quadrature levels and differences

Let $Q:\{g:[0,1]\to \mathbb{R}\}\to \mathbb{R}$ be a finite quadrature functional of the form $Q(g)={\sum }_{i=1}^m{w}_{i}g(x_i),$ where $m\ge 1$, the nodes $x_i\in [0,1]$ are distinct, and $w_i\in \mathbb{R}$. A one-dimensional nested level hierarchy is a sequence $(Q_{\mathrm{\ell }}{)}_{\mathrm{\ell }\in {\mathbb{N}}_0}$ of such functionals, with node sets $X_{\mathrm{\ell }}=\{x_{\mathrm{\ell },i}:1\le i\le m_{\mathrm{\ell }}\}$ satisfying $X_{\mathrm{\ell }-1}\subseteq X_{\mathrm{\ell }}$ for every $\mathrm{\ell }\ge 1$. Set $Q_{-1}:=0$, the zero functional, and define the detail functional ${\mathrm{\Delta }}_0:=Q_0,{\mathrm{\Delta }}_{\mathrm{\ell }}:=Q_{\mathrm{\ell }}-Q_{\mathrm{\ell }-1}(\mathrm{\ell }\ge 1)\mathrm{.}$ Finite sums are as in Finite sums and finite products, by recursion. For each $d\ge 1$, the integration functional used below is $I_d(f):={\int }_{[0,1{]}^d}f$, with the multiple Riemann integral convention of The lower and upper Darboux integrals over a nondegenerate rectangle in ${\mathbb{R}}^m$.

The convergence and rate results in this pair use the dyadic composite- Simpson hierarchy $Q_{\mathrm{\ell }}(g):=Q_S(g;0,1,2^{\mathrm{\ell }+1}),X_{\mathrm{\ell }}=\{k\mathrm{/}{2}^{\mathrm{\ell }+1}:0\le k\le 2^{\mathrm{\ell }+1}\},$ where $Q_S$ is the even-panel rule of Composite Simpson rule on an even uniform grid. The number of panels is even and at least two. The sets are nested because $k\mathrm{/}{2}^{\mathrm{\ell }}=2k\mathrm{/}{2}^{\mathrm{\ell }+1}$, and their weights are nonnegative with total weight one: each two-panel block has nonnegative weights $H(1,4,1)\mathrm{/}6$ summing to its length $H$, and the blocks partition an interval of length one. Level zero has nodes $0,1\mathrm{/}2,1$ and weights $1\mathrm{/}6,4\mathrm{/}6,1\mathrm{/}6$.

For a multi-index $\mathrm{\ell }=({\mathrm{\ell }}_1,\dots ,{\mathrm{\ell }}_d)\in {\mathbb{N}}_0^d$, any tensor product $R_{{\mathrm{\ell }}_1}^{(1)}\otimes \cdots \otimes R_{{\mathrm{\ell }}_d}^{(d)}$ of these finite functionals means the finite weighted functional obtained by multiplying the coordinate weights and evaluating on the corresponding node tuples. In particular, ${\mathrm{\Delta }}_{\mathrm{\ell }}:={\mathrm{\Delta }}_{{\mathrm{\ell }}_1}^{(1)}\otimes \cdots \otimes {\mathrm{\Delta }}_{{\mathrm{\ell }}_d}^{(d)}$. No exactness or error property is implied by the notation alone.