I would like to discuss some issues about convergence of Gaussian quadrature rules for integration. Let $I\subseteq\mathbb{R}$ be an interval (maybe infinite), $f:I\rightarrow\mathbb{R}$ be a function, and $w(x)>0$ be a function on $I$ such that $x^k w(x)\in L^1(I)$ for all $k\geq0$. Let $\{P_n\}_{n=0}^\infty$ be a sequence of orthogonal polynomials in $L_w^2(I)$ (degree of $P_n$ is $n$). A Gaussian quadrature rule approximates $$ \int_I f(x)w(x)\,dx\approx Q_n(f)=\sum_{i=1}^{n} f(x_i^{(n)})w_i^{(n)},\quad (*) $$ where $\{x_i^{(n)}\}_{i=1}^n$ are the zeros of $P_n$, and $\{w_i^{(n)}\}_{i=1}^n$ are the integration weights such that the rule is exact for polynomials of degree less than or equal to $2n-1$. The remainder of the approximation $(*)$ takes the form $C_n f^{(2n)}(\eta_n)$, where $C_n>0$ is a constant and $\eta_n\in I$.
Here are the aspects I would like to discuss about the convergence in $(*)$ as $n\rightarrow\infty$:
If $I$ is compact, then $Q_n(f)$ converges for continuous functions $f$. Let $\epsilon>0$. We take a polynomial $p$ of degree $m_\epsilon$ such that $\|f-p\|_\infty<\epsilon/(2\int_I w(x)\,dx)$ (by Stone–Weierstrass theorem). Let $n\geq m_\epsilon$. As $Q_n$ is exact for $p$, \begin{align*} \left|\int_I f(x)w(x)\,dx- Q_n(f)\right|\leq & \left|\int_I f(x)w(x)\,dx-\int_I p(x)w(x)\,dx\right| \\ + & \left|\sum_{i=1}^{n} f(x_i^{(n)})w_i^{(n)}-\sum_{i=1}^{n} p(x_i^{(n)})w_i^{(n)}\right| \\ \leq & \|f-p\|_\infty \left(\int_I w(x)\,dx+ \sum_{i=1}^{n} w_i^{(n)}\right) \\ = & \|f-p\|_\infty \left(2\int_I w(x)\,dx\right)<\epsilon. \end{align*} When $I$ is unbounded, does convergence hold for continuous functions $f$ such that $fw\in L^1(I)$?
Is the following statement true? As $f$ is smoother, faster convergence of $(*)$ holds with $n$.