Calculation of derivatives in the $L_p$ spaces where $1 \le p \le \infty$

A. N. Morozov

Modelirovanie i analiz informacionnyh sistem, Tome 27 (2020) no. 1, pp. 124-131.

Résumé

It is well known in functional analysis that construction of $k$-order derivative in Sobolev space $W_p^k$ can be performed by spreading the $k$-multiple differentiation operator from the space $C^k.$ At the same time there is a definition of $(k,p)$-differentiability of a function at an individual point based on the corresponding order of infinitesimal difference between the function and the approximating algebraic polynomial $k$-th degree in the neighborhood of this point on the norm of the space $L_p$. The purpose of this article is to study the consistency of the operator and local derivative constructions and their direct calculation. The function $f\in L_p[I]$, $p>0$, (for $p=\infty$, we consider measurable functions bounded on the segment $I$ ) is called $(k; p)$-differentiable at a point $x \in I$ if there exists an algebraic polynomial of $\pi$ of degree no more than $k$ for which holds $ \Vert f-\pi \Vert_{L_p[J_h]} = o(h^{k+\frac{1}{p}}), $ where $J_h=[x_0-h; x_0+h]\cap I.$ At an internal point for $k = 1$ and $p = \infty$ this is equivalent to the usual definition of the function differentiability. The discussed concept was investigated and applied in the works of S. N. Bernshtein [1], A. P. Calderon and A. Sigmund [2]. The author's article [3] shows that uniform $(k, p)$-differentiability of a function on the segment $I$ for some $p\ge 1$ is equivalent to belonging the function to the space $C^k[I]$ (existence of an equivalent function in $C^k[I]$). In present article, integral-difference expressions are constructed for calculating generalized local derivatives of natural order in the space $L_1$ (hence, in the spaces $L_p$, $1\le p\le \infty$), and on their basis — sequences of piecewise constant functions subordinate to uniform partitions of the segment $I$. It is shown that for the function $ f $ from the space $ W_p^k $ the sequence piecewise constant functions defined by integral-difference $k$-th order expressions converges to $ f^{(k)} $ on the norm of the space $ L_p[I].$ The constructions are algorithmic in nature and can be applied in numerical computer research of various differential models.

Détail

Keywords: differentiability of function in the spaces $L_p$, differences for the space $L_1$, numerical finding of derivatives on a computer, the spreading of the differentiation operator.

Bibliographie

[1] S. Bernstein, “On the question of local best approximation of functions”, Dokl. USSR Acad. Sci., 26:9 (1940), 839–842

[2] A. Calderón, A. Zygmund, “Local properties of solutions of elliptic partial differential equations”, Selected papers of Antoni Zygmund, Springer, 1989, 285–339 | MR

[3] A. Morozov, “On Taylor differentiability in the spaces $L_p,$ $0

\leq\infty$”, Modeling and analysis of inform. systems, 25:3 (2018), 323–330 | MR

[4] A. Morozov, “Local approximations of differentiable functions”, Math. Notes, 100:2 (2016), 256–262 | MR | Zbl

[5] V. Abilov, F. Abilova, “Problems”, Math. Notes, 76 (2004), 749–757 | MR | Zbl

[6] A. Khromov, G. Khromova, “Discontinuous steklov operators in the problem of uniform approximation of derivatives on an interval”, Computational Mathematics and Mathematical Physics, 54:9 (2014), 1389–1394 | MR | Zbl

[7] V. Dzyadyk, Introduction to the theory of uniform approximation of functions by polynomials, Nauka, 1977 | MR | Zbl

[8] L. Schumaker, Spline functions: basic theory, Wiley, New York, 1981 | MR | Zbl