Using Hermite-Gauss Quadrates to approximate the integral $I = \int_0^\infty e^{ -x^2} f(x) \, dx$, the error is given as $$E = \frac{m!\sqrt{\pi}}{2^m (2m)!} f^{(2m)}(\theta)$$ with $0 < \theta < \infty$. However, this estimation is not practical since in some cases we cant know $f^{(2m)}(\theta)$. So are there any ways to determine that error or bounds of that one?
Many times we know that $|f^{(2 m)}(\theta)|$ is bounded. For example, $f(x) = \sin{(a x)}$ implies that $|f^{(2 m)}(\theta)| < a^{2 m}$. In this case, the error is bounded and we may say that the error in using Hermite-Gauss quadrature to this order is less than some number, which is valuable.