Numerical modeling tools and $S$-derivatives

Numerical study of various processes leads to the need of clarification (extensions) of the limits of applicability of computational constructs and modeling tools. For dynamical systems, this question may be related with a generalization of the concept of a derivative, which keeps the used constructions relevant. In this article we introduce the concept of weak local differentiability in a space of Lebesgue integrable functions and consider the consistency of this concept with such fundamental computational constructions as the Taylor expansion and finite differences, as well as properties of functions with a given type of differentiability on a segment.
The function $f$ from $L_1[a; b]$ is called $S$-differentiable at the point $x_0$ from $(a; b)$, if there are coefficients $c$ and $q$, for which $\int_{x_0}^{x_0+h}(f(x)-c-q\cdot(x-x_0))dx=o(h^2)$. Formulas are found for calculating the coefficients $c$ and $q$, coefficients $c$ and $q$, which are conveniently denoted $f_S(x_0)$ and $f'_S(x_0)$ respectively. It is shown that if the function $f$ belongs to $W_1^{n-1}[a;b]$, $n$ is greater than $1$, and the function $f^{(n-1)}$ is $S$-differentiable at the point $x_0$ from $(a; b)$, then $f$ is approximated by a Taylor polynomial with accuracy $o((x-x_0)^n)$, and the ratio of , а отношение $\Delta_h^n(f,x_0)$ to $h^n$ tends to $f_s^{(n)}(x_0)$ as $h$ tends to $0$. Based on the quotient $\Delta_h^n(f;\cdot)$ and $h^n$, a sequence is built $\{\Lambda_m^n[f]\}$ piecewise constant functions subordinate to partitions segment $[a; b]$ into $m$ equal parts. It is shown that for the function $f$ from $W_1^{n-1}[a;b]$, for which the value is defined $f_s^{(n)}(x_0)$, $\{\Lambda_m^n[f](x_0)\}$ converges to $f_s^{(n)}(x_0)$ as $m$ tends to infinity, and for $f$ from $W_p^n[a;b]$ the sequence $\{\Lambda_m^n[f]\}$ converges to $f^{(n)}$ in the norm of the space $L_p[I]$. The place of $S$-differentiability in practical and theoretical terms is determined by its bilateral relations with ordinary differentiability. It is proved that if $f$ belongs to $W_1^{n-1}[I]$ and the function $f^{(n-1)}$ is uniformly $S$-differentiable on $I$, then $f$ belongs to $C^n[I]$. The constructions are algorithmic in nature and can be applied in numerically computer research of various relevant models.