Calculation of Derivatives in the Lp Spaces where 1 ≤ p ≤ ∞
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 i...
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Yaroslavl State University
2020-03-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/1293 |
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| author | Anatoly Nikolaevich Morozov |
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| description | 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. |
| format | Article |
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| issn | 1818-1015 2313-5417 |
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| spelling | doaj-art-fd34b34a0d044f10bd52e18ea2c073c42025-08-20T03:00:45ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172020-03-0127112413110.18255/1818-1015-2020-1-124-131964Calculation of Derivatives in the Lp Spaces where 1 ≤ p ≤ ∞Anatoly Nikolaevich Morozov0P. G. Demidov Yaroslavl State UniversityIt 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.https://www.mais-journal.ru/jour/article/view/1293differentiability of function in the spaces lpdifferences for the space l1numerical finding of derivatives on a computerthe spreading of the differentiation operator |
| spellingShingle | Anatoly Nikolaevich Morozov Calculation of Derivatives in the Lp Spaces where 1 ≤ p ≤ ∞ Моделирование и анализ информационных систем differentiability of function in the spaces lp differences for the space l1 numerical finding of derivatives on a computer the spreading of the differentiation operator |
| title | Calculation of Derivatives in the Lp Spaces where 1 ≤ p ≤ ∞ |
| title_full | Calculation of Derivatives in the Lp Spaces where 1 ≤ p ≤ ∞ |
| title_fullStr | Calculation of Derivatives in the Lp Spaces where 1 ≤ p ≤ ∞ |
| title_full_unstemmed | Calculation of Derivatives in the Lp Spaces where 1 ≤ p ≤ ∞ |
| title_short | Calculation of Derivatives in the Lp Spaces where 1 ≤ p ≤ ∞ |
| title_sort | calculation of derivatives in the lp spaces where 1 ≤ p ≤ ∞ |
| topic | differentiability of function in the spaces lp differences for the space l1 numerical finding of derivatives on a computer the spreading of the differentiation operator |
| url | https://www.mais-journal.ru/jour/article/view/1293 |
| work_keys_str_mv | AT anatolynikolaevichmorozov calculationofderivativesinthelpspaceswhere1p |