On Computational Constructions in Function Spaces

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

Difference Expressions, Integral Averaging, Continuity Operator, Taylor Polynomial, Numerical Finding of Derivatives on a Computer

1. A. P. Calderon and A. Zygmund, “Local Properties of Solution of Elliptic Partial Differential Equation,” Studia Mathematica, vol. 20, no. 2, pp. 171–225, 1961.

2. A. N. Morozov, “Numerical Modeling Tools and S-Derivatives,” Modeling and analysis of inform. systems, vol. 29, no. 1, pp. 20–29, 2022.

3. A. N. Morozov, “Calculation of Derivatives in the $L_p$ Spaces where $1leq pleq infty,$,” Modeling and analysis of inform. systems, vol. 27, no. 1, pp. 124–131, 2020.

4. V. K. Dzyadyk, Introduction to the Theory of Uniform Approximation of Functions by Polynomials. Nauka, 1977.

5. L. L. Schumaker, Spline Functions: Basic Theory. Wiley, New York, 1981.

6. V. I. Bogachev, Measure Theory. V.1. Springer, 2007.

7. L. V. Kantorovich and G. P. Akilov, Functional Analysis. Nauka, Moscow, 1984.

8. Y. A. Brudnyi, “Criteria for the Existence of Derivatives in Lp,” Mathematics of the USSR-Sbornik, vol. 2, no. 1, pp. 35–55, 1967.