On the Taylor differentiability in spaces $L_p, 0$

A. N. Morozov

Modelirovanie i analiz informacionnyh sistem, Tome 25 (2018) no. 3, pp. 323-330

Résumé

The function $f\in L_p[I], \;p>0,$ is called $(k,p)$-differentiable at a point $x_0\in I$ 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. At an interior point for $k=1$ and $p=\infty$, the definition is equivalent to the usual differentiability of the function. There is a standard "hierarchy" for the existence of differentials(if $p_1$ then $(k,p_2)$-differentiability should be $(k,p_1)$-differentiability.) In the works of S.N. Bernstein, A.P. Calderon and A. Zygmund were given applications of such a construction to build a description of functional spaces ($p=\infty$) and the study of local properties of solutions of differential equations $(1\le p\le\infty)$, respectively. This article is related to the first mentioned work. The article introduces the concept of uniform differentiability. We say that a function $f$, $(k,p)$-differentiable at all points of the segment $I$, is uniformly $(k,p)$-differentiable on $I$ if for any number $\varepsilon>0$ there is a number $\delta>0$ such that for each point $x\in I$ runs $ \Vert f-\pi\Vert_{L_p[J_h]}\varepsilon\cdot h^{k+\frac{1}{p}} \; $ for $0$ where $\pi$ is the polynomial of the terms of the $(k, p)$-differentiability at the point $x$. Based on the methods of local approximations of functions by algebraic polynomials it is shown that a uniform $(k,p)$-differentiability of the function $f$ at some $1\le p\le\infty$ implies $f\in C^k[I].$ Therefore, in this case the differentials are "equivalent". Since every function from $C^k[I]$ is uniformly $(k,p)$-differentiable on the interval $I$ at $1\le p\le\infty,$ we obtain a certain criterion of belonging to this space. The range $0$ obviously, can be included into the necessary condition the membership of the function $C^k[I]$, but the sufficiency of Taylor differentiability in this range has not yet been fully proven.

Détail

Keywords: Taylor differentiability of function, local approximations of functions.