Countable Additivity of Spreading the Differentiation Operator
In this article, we continue the study of the properties acquired by the differentiation operator Λ with spreading beyond the space W1¹. The study is conducted by introducing the family of spaces Yp¹ , 0 < p < 1, having analogy with the family Wp¹, 1 ≤ p < ∞. Spaces Yp¹ are equi...
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Yaroslavl State University
2014-06-01
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| Series: | Моделирование и анализ информационных систем |
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| Online Access: | https://www.mais-journal.ru/jour/article/view/111 |
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| author | A. N. Morozov |
| author_facet | A. N. Morozov |
| author_sort | A. N. Morozov |
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| description | In this article, we continue the study of the properties acquired by the differentiation operator Λ with spreading beyond the space W1¹. The study is conducted by introducing the family of spaces Yp¹ , 0 < p < 1, having analogy with the family Wp¹, 1 ≤ p < ∞. Spaces Yp¹ are equiped with quasinorms constructed on quasinorms spaces Lp as the basis; Λ : Yp¹ → Lp. We have given a sufficient condition for a function, piecewise belonging to the space Yp¹ to be in this space (if f ∈ Yp¹ [xi-1; xi ], i ∈ N, 0 = x0 < x1 < · · · < xi < · · · < 1, then f ∈ Yp¹ [0; 1]). In other words, it is the sign when the equality: Λ(S fi) = S Λ(fi) is true. The bounded variation in the Jordan sense is closest to the sufficient condition among the classic characteristics of functions. As a corollary, it comes out that, if a function f piecewise belongs to the space of W1¹ and has a bounded variation, f belongs to each space Yp¹ , 0 < p < 1. |
| format | Article |
| id | doaj-art-c4359f5846c94e34a341af6ed64288b3 |
| institution | Matheson Library |
| issn | 1818-1015 2313-5417 |
| language | English |
| publishDate | 2014-06-01 |
| publisher | Yaroslavl State University |
| record_format | Article |
| series | Моделирование и анализ информационных систем |
| spelling | doaj-art-c4359f5846c94e34a341af6ed64288b32025-08-04T14:06:40ZengYaroslavl State UniversityМоделирование и анализ информационных систем1818-10152313-54172014-06-01213819010.18255/1818-1015-2014-3-81-90105Countable Additivity of Spreading the Differentiation OperatorA. N. Morozov0P.G. Demidov Yaroslavl State UniversityIn this article, we continue the study of the properties acquired by the differentiation operator Λ with spreading beyond the space W1¹. The study is conducted by introducing the family of spaces Yp¹ , 0 < p < 1, having analogy with the family Wp¹, 1 ≤ p < ∞. Spaces Yp¹ are equiped with quasinorms constructed on quasinorms spaces Lp as the basis; Λ : Yp¹ → Lp. We have given a sufficient condition for a function, piecewise belonging to the space Yp¹ to be in this space (if f ∈ Yp¹ [xi-1; xi ], i ∈ N, 0 = x0 < x1 < · · · < xi < · · · < 1, then f ∈ Yp¹ [0; 1]). In other words, it is the sign when the equality: Λ(S fi) = S Λ(fi) is true. The bounded variation in the Jordan sense is closest to the sufficient condition among the classic characteristics of functions. As a corollary, it comes out that, if a function f piecewise belongs to the space of W1¹ and has a bounded variation, f belongs to each space Yp¹ , 0 < p < 1.https://www.mais-journal.ru/jour/article/view/111differentiation operatorquasinorm |
| spellingShingle | A. N. Morozov Countable Additivity of Spreading the Differentiation Operator Моделирование и анализ информационных систем differentiation operator quasinorm |
| title | Countable Additivity of Spreading the Differentiation Operator |
| title_full | Countable Additivity of Spreading the Differentiation Operator |
| title_fullStr | Countable Additivity of Spreading the Differentiation Operator |
| title_full_unstemmed | Countable Additivity of Spreading the Differentiation Operator |
| title_short | Countable Additivity of Spreading the Differentiation Operator |
| title_sort | countable additivity of spreading the differentiation operator |
| topic | differentiation operator quasinorm |
| url | https://www.mais-journal.ru/jour/article/view/111 |
| work_keys_str_mv | AT anmorozov countableadditivityofspreadingthedifferentiationoperator |