Lets say that we have a new Quadrature rule $Q^{(1)}_n(f)$ with degree of exactness $n-1$ and we have managed to express it's error term $R^{(1)}_n(f)$ via other already known quadrature rules. For example, I have $$R^{(1)}_n(f)=R^{(G,1)}_n(f) +R^{(F,1)}_n(f),$$ where $R^{(G,1)}_n(f)$ is the error term of the well known Gauss-Chebyshev quadrature rule of the first kind having degree of exactness $2n-1$ and $R^{(F,1)}_n(f)$ is the Fejer quadrature rule of the first kind with degree of exactness $n-1$. Also it is well known that we have:
$R^{(G,1)}_n(f)=\frac{\pi}{2^{2n-1}}\frac{f^{(2n)}(\xi)}{(2n)!}$ for $f \in C^{(2n)}[-1,1]$
For $f \in C^{(n)}[-1,1]$, using the Peano Kernel Theorem, we have $$|R^{(F,1)}_n(f)| \leq c^{(F,1)}_{n} \cdot \frac{max |f^{(n)}|}{n!}$$ and it is well known that the best possible relation for $c^{(F,1)}_{n}$ as $n \to \infty$ ($\textbf{asymptotically}$) is $O(2^{-n} n^{-2})$
Now, for the new quadrature rule $Q^{(1)}_n(f)$, for $f \in C^{(n)}[-1,1]$, using the Peano Kernel Theorem, we have $$|R^{(1)}_n(f)| \leq c^{(1)}_{n} \cdot \frac{max |f^{(n)}|}{n!}.$$ Can we have a result about the best possible relation for $c^{(1)}_{n}$ as $n \to \infty$ ($\textbf{asymptotically}$), using 1) and/or 2)? In order to do that, should we know the asymptotic best constant for $R^{(G,1)}_n(f)$ for $f \in C^{(n)}[-1,1]$?