Calculation of Derivatives in the Lp Spaces where 1 ≤ p ≤ ∞

Abstract

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.