An approximation of integrable functions by step functions with an application

Abstract:

Let $f \in {L^1}(0,\infty ),\delta > 0$ and $({G_\delta }f)(t) = {\delta ^{ - 1}}\smallint _t^\infty {e^{(t - s)/\delta }}f(s)ds$. Given a partition $P = \{ 0 = {t_0} < {t_1} < \cdots < {t_i} < {t_{i + 1}} < \cdots \}$ of $[0,\infty )$ where ${t_i} \to \infty$, we approximate f by the step function ${A_P}f$ defined by \[ {A_P}f(t) = ({G_{{\delta _i}}}{G_{{\delta _{i - 1}}}} \cdots {G_{{\delta _i}}}f)(0)\quad {\text {for}}\;{t_{i - 1}} \leqslant t < {t_i},\] where ${\delta _i} = {t_i} - {t_{i - 1}}$. The main results concern several properties of this process, with the most important one being that ${A_P}f \to f$ in ${L^1}(0,\infty )$ as $\mu (P) = {\sup _i}{\delta _i} \to 0$. An application to difference approximations of evolution problems is sketched.