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.
