On Computational Constructions in Function Spaces

Numerical study of various processes leads to the need for clarification (extensions) limits applicability computational constructs and modeling tools. In this article, we differentiability in space Lebesgue integrable functions consistency concept with fundamental constructions such as Taylor expansion finite differences is considered. The function $f$ from $L_1 [a; b]$ called $(k,L)$-differentiable at point $ x_0$ $(a; b),$ if there exists an algebraic polynomial $P,$ degree no higher than $k,$ that integral over segment ${x_0}$ then ${x_0+h}$ $f-P$ $o(h^{k+1}).$ Formulas are found calculating coefficients representing limit ratio modifications {\bf\Delta}_h^m(f,x) h^m\!, \; m=1, \cdots, k. It turns out $f\!\in\!W_1^{l}[a; b],$ $f^{(l)}$ $x_0,$ approximated by a up o\big((x{-}x_0)^{l+k}\big),$ can be above way. To $L_1$ on set, discrete "global" construction difference expression used: based quotient ${\bf\Delta}_h^m(f, \cdot)$ $h^m$ sequence built $\big{{\bf\Lambda}_n^m[f]\big}$ piecewise constant subordinate partitions half-interval $[a; b)$ into $n$ equal parts. shown $x_0$ $\big{{\bf\Lambda}_n^m[f]\big},\; m=1,\cdots, k, converge $n\to \infty$ approximating it. Using $\big{{\bf\Lambda}_n^k[f]\big}$ following theorem established: {\it "$f$ $L_1[a;b]$ belongs $C^k[a;b] \Longleftrightarrow uniformly $[a;b]$".} A special place occupied corresponding case $m\!=\!0.$ We consider them $L_1[Q_0],$ where $Q_0$ cube $\mathbb R^d.$ Given f\!\in\!L_1$ partition $\tau_{n}$ semi-closed $\;n^d$ cubes construct $\Theta_n[f]$, defined average each $Q\!\in\!\tau_{n}.$ This theoretical facts: 1)$f$ $L_p, 1 \le p < \infty, \big{\Theta_n[f] \big}$ converges $L_p;$ boundedness $\big{\Theta_n[f]\big} f\!\in\!L_\infty;$ 2) sequences $\big{\Theta_n[\cdot]\big}$ define equivalence classes operator-projector $\Theta$ $L_1;$ 3) $f\!\in\!L_{\infty}$ get \overline{\Theta [f]}\!\in\!B,$ $B$ bounded functions, [f]}$ \Theta [f](x),$ extended set measure zero equality $\;\big\Vert [f]}\big\Vert_{B} = \Vert f\Vert_{\infty}.$} Thus, family spaces $L_p$ one replace $L_{\infty}[Q_0]$ $B[Q_0].$