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(i) Your purported error bound is of course incorrect: consider e.g. $f(t)=t$. (ii) To get rid of the singularity, make the substitution $u=\sqrt t$, so that $$\int_0^T dt\,f(t)=\int_0^T \frac{dt}{\sq …

$\newcommand\Z{\mathbb Z}\newcommand\R{\mathbb R}\newcommand\la{\lambda}$If $f$ is an arbitrary Lebesgue-integrable function, then, as is done in a definition of the Lebesgue integral, it makes sense …

$\newcommand\erf{\operatorname{erf}}\newcommand\R{\mathbb R}$The functional equation in question is \begin{equation*} a=F(a) \tag{1}\label{1} \end{equation*} on $(0,\infty)$, where $a$ is in the c …

The key here is the simple change-of-interval/rescaling formula, found e.g. at the link in the OP, according to which \begin{equation} T_n(f)(x)=T_{n,[0,x]}(f)=x\sum_1^n w_i f(xx_i), \tag{*} \end{eq …