Preview

Modeling and Analysis of Information Systems

Advanced search

Completion of the Kernel of the Differentiation Operator

https://doi.org/10.18255/1818-1015-2017-1-111-120

Abstract

When investigating piecewise polynomial approximations in spaces \(L_p, \; 0~<~p~<~1,\) the author considered the spreading of k-th derivative (of the operator) from Sobolev spaces \(W_1 ^ k\) on spaces that are, in a sense, their successors with a low index less than one. In this article, we continue the study of the properties acquired by the differentiation operator \(\Lambda\) with spreading beyond the space \(W_1^1\)
$$
\Lambda~:~W_1^1~\mapsto~L_1,\; \Lambda f = f^{\;'}
$$.
The study is conducted by introducing the family of spaces \(Y_p^1, \; 0 <p < 1,\) which have analogy with the family \(W_p^1, \; 1 \le p <\infty.\) This approach gives a new perspective for the properties of the derivative. It has been shown, for example, the additivity property relative to the interval of the spreading differentiation operator:
$$ \bigcup_{n=1}^{m} \Lambda (f_n) = \Lambda (\bigcup_{n=1}^{m} f_n).$$
Here, for a function \(f_n\) defined on \([x_{n-1}; x_n], \; a~=~x_0 < x_1 < \cdots <x_m~=~b\), \(\Lambda (f_n)\) was defined. One of the most important characteristics of a linear operator is the composition of the kernel.
During the spreading of the differentiation operator from the space \( C ^ 1 \) on the space \( W_p ^ 1 \) the kernel does not change. In the article, it is constructively shown that jump functions and singular functions \(f\) belong to all spaces \( Y_p ^ 1 \) and \(\Lambda f = 0.\) Consequently, the space of the functions of the bounded variation \(H_1 ^ 1 \) is contained in each \( Y_p ^ 1 ,\) and the differentiation operator on \(H_1^1\) satisfies the relation \(\Lambda f = f^{\; '}.\)
Also, we come to the conclusion that every function from the added part of the kernel can be logically named singular.

About the Author

Anatoly N. Morozov
P.G. Demidov Yaroslavl State University
Russian Federation

PhD,

14 Sovetskaya str., Yaroslavl 150003



References

1. English transl.: Morozov A. N., “Local Approximations of Differentiable Functions”, Math. Notes, 100:2 (2016), 256–262.

2. Morozov A. N., “Kusochno-polinomialnye priblizheniay i differentsiruemost v prostranstvakh Lp (0 < p < 1)”, Modeling and analysis of inform. systems, 12:1 (2005), 18–21, (in Russian).

3. Morozov A. N., “Countable Additivity of spread of the Differentiation Operator”, Modeling and analysis of inform. systems, 21:3 (2014), 81–90, (in Russian).

4. Morozov A. N., “On Smoothness in Lp (0 < p < 1)”, Modeling and analysis of inform. systems, 19:3 (2012), 97–104, (in Russian).

5. Kantorovich L. V., Akilov G. P., Functional analysis, ed. Howard L. Silcock, Pergamon Press, Oxford, New York, 1982.

6. Berg J., Lofsrom J., Interpolation Spaces. An Introduction, Springer-Verlag, 1976.

7. Smolin U. N., Vvedenie v teoriyu funktsy deistvitelnoi peremennoi, FLINTA, M., 2012, (in Russian).

8. Timan A. F., Theory of Approximation of Functions of a Real Variable, Courier Dover Publications, 1994.


Review

For citations:


Morozov A.N. Completion of the Kernel of the Differentiation Operator. Modeling and Analysis of Information Systems. 2017;24(1):111-120. (In Russ.) https://doi.org/10.18255/1818-1015-2017-1-111-120

Views: 882


Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.


ISSN 1818-1015 (Print)
ISSN 2313-5417 (Online)