Composite Simpson error bound for Chebyshev weighted mass entries

Given: Integers $N\ge 0$, even $M\ge 2$, $h=\pi \mathrm{/}M$, and indices $0\le i,j\le N$.

Arcsine-weighted inner product on the Chebyshev interval: $⟨T_i,T_j{⟩}_C={\int }_0^{\pi }{T}_i(\cos \theta )T_j(\cos \theta )d\theta$.

Composite-Simpson assembly of the Chebyshev weighted mass matrix: $M_{ij}^S$ is the composite Simpson value on $[0,\pi ]$ for $g_{ij}(\theta )=T_i(\cos \theta )T_j(\cos \theta )$ with step $h$.

Composite Simpson error bound for a four-times continuously differentiable function: if $g\in C^4([0,\pi ])$, then $\mathrm{\mid }{\int }_0^{\pi }g-Q_S(g;0,\pi ,M)\mathrm{\mid }\le \pi h^4\max \mathrm{\mid }{g}^{(4)}\mathrm{\mid }\mathrm{/}180$.

$T_n(\cos \theta )=\cos (n\theta )$ and $U_n(\cos \theta )\sin \theta =\sin ((n+1)\theta )$ for every $n\in \mathbb{N}$ and The addition formulas for sine and cosine: $T_k(\cos \theta )=\cos (k\theta )$ and $2\cos u\cos v=\cos (u+v)+\cos (u-v)$.

The derivatives of sine and cosine are cosine and minus sine and The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$ give the derivatives of each $\cos (k\theta )$; Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$ distributes these rules over the finite sum in step 1.1. The resulting derivatives are finite linear combinations of sine and cosine functions, which are continuous by differentiability and A function differentiable at $c$ is continuous at $c$; Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function preserves continuity under those sums. Higher derivatives and the classes $C^k$ and $C^{\mathrm{\infty }}$ is the $C^4$ convention.