The $n$-th Gauss-Laguerre quadrature scheme aims to approximate integral of exponentially decreassing function over $[0;+ \infty[$ by a finite sum, according to: $ \displaystyle { \int _0 ^{+ \infty} f(x) \, dx \approx \sum _{i = 1} ^n \omega_i f(x_i) }$, where $x_1$, $\cdots$, $x_n$ are the roots of the $n$-th Laguerre polynomial $L_n$ and the weights $\omega_1$, $\cdots$, $\omega_n$ are defined by $\omega_i = \dfrac{1}{x_i(L'_n(x_i))^2}$, $1 \leq i \leq n$.
I am interested in this quadrature to apply it to bounded $\mathcal{C}^\infty$-functions over $[0 ; + \infty [$, with bounded derivatives. In particular, I want to understand the error $\varepsilon_n(f)$ produced by the $n$-th degree Gauss-Laguerre quadrature.
For this purpose, we know three facts:
The Uspensky theorem states that $ \varepsilon_n(f) \underset{n \longrightarrow + \infty}{\longrightarrow} 0 $ for functions $f$ satisfying $|f(x)| \leq c x^{-(1 + \rho)} e^x$ for large $x >> 1$ and some $\rho > 0$. (See [1])
A $\mathcal{C}^\infty$-function $f$ is an element of $C_q^p[0,\infty)$ for all integers $p$ and $q$ such that $0 \leq p \leq q$ (for the notations, see here and here). According to Mastroianni-Monegati theorem, we have for $q = 2p + 3$: \begin{equation} \varepsilon_n(f) \underset {n \longrightarrow + \infty} {=} \mathcal{O} \left ( \dfrac{1}{n^{p - 1}} \right ) E_{n - p - 1}(\Phi, e^{-\frac{x}{2}}) \end{equation} where $\Phi(x) := x^{p + 3} f(x) \in \mathcal{C}^q([0, + \infty[)$ and $E_n(f ; \omega) := \underset{p_n \in \mathbb{R}_n[X]}{\text{inf}} ||\omega (f - p_n)||_{\infty, [0; + \infty[} $ denotes the best weighted polynomial approximation, the infimum being taken over all polynomial of degree at most $n$. See [2].
Moreover, if the $\mathcal{C}^\infty$ function $f$ is such that for all positive integer $n$, $||f^{(n)}(x) x^{\frac{n}{2}} e^{-x}||_{\infty, [0;+ \infty[} < + \infty$, then $f$ belongs to each Sobolev-type space $W_\infty^n(e^{-x})$, $n \geq 1$, defined by \begin{equation} W_\infty^n(e^{-x}) = \left \{ f \in L^\infty_{e^{-x}} \,,\, f^{(n - 1)} \text{ abs. continuous on }(0, \infty) \text{ and } ||f^{(n)}(x) x^{\frac{n}{2}} e^{-x}||_{\infty, [0;+ \infty[} < + \infty \right \} \end{equation} where \begin{equation} L^\infty_{e^{-x}} = \left \{ f \in \mathcal{C}([0 ; + \infty[) \,,\, \underset{x \longrightarrow + \infty}{\text{lim}} f(x) e^{-x} = 0 \right \} \ . \end{equation} Consequently, $E_n(f ; \omega) \leq \dfrac{C}{\sqrt{n}^r}$ according to [3], prop. 4.1. p. 112.
This is the case for example if $f$ is bounded, as well as all its derivatives.
Alltogether allows us to deduce that for any bounded $\mathcal{C}^\infty$-function $f$ with bounded derivative, we have $\varepsilon_n(f) \underset{n \longrightarrow + \infty}{=} \mathcal{O}\left ( \dfrac{1}{n^{\frac{k}{2}}} \right )$ for all positive integer $k$.
My question is therefore :
Do we have exact estimation of the speedness of the convergence for bounded functions $f: \mathbb{R}^+ \longrightarrow \mathbb{R}$ with infinite regularity and bounded derivatives?
As an exemple, we can consider $f_0(x) = \displaystyle { \dfrac {1} {1 + x} }$. Easily computer computations give: \begin{equation} \begin{array}{|c|l|l|} \hline \text{Degree } n & \text{Error } \varepsilon_n(f_0)& -\ln(\varepsilon_n(f_0) \\ \hline 1 & 9.634 \cdot 10^{-2} & 2,339 \\ \hline 2 & 2.491 \cdot 10^{-2} & 3,692 \\ \hline 4 & 3.045 \cdot 10^{-3} & 5,793 \\ \hline 8 & 1.326 \cdot 10^{-4} & 8,927 \\ \hline 16 & 1.394 \cdot 10^{-6} & 13,483 \\ \hline 32 & 2.025 \cdot 10^{-9} & 20,017 \\ \hline 64 & 1.840 \cdot 10^{-13} & 29,323 \\ \hline 128 & 3.380 \cdot 10^{-19} & 42,531 \\ \hline 256 & 2.526 \cdot 10^{-27} & 61,242 \\ \hline 512 & 7.940 \cdot 10^{-39} & 87,728 \\ \hline 1024 & 4.219 \cdot 10^{-55} & 125,202 \\ \hline 2048 & 4.021 \cdot 10^{-78} & 178,209 \\ \hline 4096 & 1.107 \cdot 10^{-110} & 253,182 \\ \hline 8192 & 9.882 \cdot 10^{-157} & 359,215 \\ \hline \end{array} \end{equation}
From these computations, one can conjecture that $\varepsilon_{2^n}(f_0) \underset{n \longrightarrow + \infty}{\sim} \exp \left ( -4 \sqrt{2}^n + 2 \sqrt{2} \right )$, from which one can conjecture: \begin{equation} \varepsilon_n(f_0) \underset{n \longrightarrow + \infty}{\sim} \exp \left ( -4 \sqrt{n} + 2 \sqrt{2} \right ) \ . \end{equation}
References:
[1] J. V. Uspenksy, On the convergence of quadrature formulas related to an infinite interval, Trans. Amer. Math. Soc. 30 (1928), 542-559
[2] Mastroianni G., Monegato G., Convergence of product integration rules over $(0, \infty)$ for functions with weak singularities at the origin. Math. Comp. 64, (1995), 237-249.
[3] G. Mastroianni, J. Szabados, Polynomial approximation on the real semiaxis with generalized Laguerre weights. Stud. Univ. Babes-Bolyai Math. 52 (2007), nr.4, 105--128.
