On the Taylor Differentiability in Spaces Lp, 0 < p ≤ ∞

Abstract

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 \(p_1<p_2,\) 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 \(0<h<\delta, \; J_h = [x\!-\!H; x\!+\!h]\cap I,\) 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 \(0<p<1,\) 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.