The compactness principle and Vitaliʼs theorem for h-holomorphic functions
https://doi.org/10.29235/1561-2430-2022-58-4-381-388
Abstract
In this paper, we consider the properties of uniformly convergent sequences of h-holomorphic functions on the set of h-complex numbers. Theorems on the global antiderivative and on the uniform approximation of h-holomorphic functions by polynomials are formulated and proven. The sufficient conditions for the h-holomorphism of the limit function are obtained. The compactness principle for functions of an h-complex variable and an analog of Vitaliʼs theorem for h-analytic functions are formulated and proven.
About the Author
V. A. Pavlovsky
Belarusian State University
Russian Federation
Russian Federation
Vladislav A. Pavlovsky – Postgraduate Student of the Department of Function Theory, Belarusian State University.
4, Nezavisimosti Ave., 220030, Minsk
References
1. Yaglom I. M. Complex Numbers in Geometry. Moscow, Editorial URSS Publ., 2004. 192 p. (in Russian).
2. Antonuccio F. Semi-Complex Analysis and Mathematical Physics. Arxiv [Preprint], 2008. Available at: https://arxiv.org/pdf/gr-qc/9311032.pdf
3. Khrennikov A., Segre G. An Introduction to Hyperbolic Analysis. Arxiv [Preprint], 2008. Available at: https://arxiv.org/abs/math-ph/0507053. https://doi.org/10.48550/arXiv.math-ph/0507053
4. Pavlovsky V. A., Vasiliev I. L. On the properties of h-differentiable functions. Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika = Journal of the Belarusian State University. Mathematics and Informatics, 2021, no. 2, pp. 29–37 (in Russian). https://doi.org/10.33581/2520-6508-2021-2-29-37
5. Zverovich E. I. Real and Complex Analysis. Part 5. Multiple Integrals. Integrals Over Manifolds. Minsk, Vysheishaya shkola Publ., 2007. 195 p. (in Russian).
6. Zverovich E. I. Real and Complex Analysis. Part 4. Functional Sequences and Series. Integrals Depending on a Parameter. Minsk, Vysheishaya shkola Publ., 2008. 165 p. (in Russian).
7. Vasil’ev I. L., Pavlovskii V. A. Mappings with the help of h-holomorphic functions. Vestsі BDPU. Seriya 3. Fіzіka. Matematyka. Іnfarmatyka. Bіyalogіya. Geagrafіya [BGPU Bulletin. Series 3. Physics. Mathematics. Informatics. Biology. Geography], 2021, no. 2, pp. 37−43 (in Russian).
8. Pavlovsky, V. A. Differentiation and integration of functions of an h-complex variable. Nauka i obrazovaniye v sovremennom mire: Vyzovy XXI veka. Materialy IX Mezhdunarodnoy nauchno-prakticheskoy konferentsii, 15 sentyabrya 2021 [Science and Education in the Modern World: Challenges of the XXI Century. Materials of the IX International Scientific and Practical Conference, September 15, 2021]. Nur-Sultan, 2021, pp. 70–73 (in Russian).
9. Stoilow S. Theoria funcţiilr de o variabilă compexă. Volume 1. Noţiunişi principii fundamentale. Editura academiei republicii populare române, 1954. 360 p. (in Romanian).
10. Shabat B. V. Introduction to Complex Analysis. Tutorial. Part 1. Functions of One Variable. Moscow, Lenand Publ., 2015. 572 p. (in Russian).
Review
Views: 273
Email this article 